Deformations of Dimer Models
The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other h...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2022 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2022
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211638 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859726405177180160 |
|---|---|
| author | Higashitani, Akihiro Nakajima, Yusuke |
| author_facet | Higashitani, Akihiro Nakajima, Yusuke |
| citation_txt | Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon , there exists a dimer model having as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.
|
| first_indexed | 2026-03-15T11:15:08Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 030, 53 pages
Deformations of Dimer Models
Akihiro HIGASHITANI a and Yusuke NAKAJIMA b
a) Department of Pure and Applied Mathematics, Graduate School of Information Science
and Technology, Osaka University, Osaka 565-0871, Japan
E-mail: higashitani@ist.osaka-u.ac.jp
b) Department of Mathematics, Kyoto Sangyo University,
Motoyama, Kamigamo, Kita-Ku, Kyoto, 603-8555, Japan
E-mail: ynakaji@cc.kyoto-su.ac.jp
Received August 06, 2021, in final form April 10, 2022; Published online April 16, 2022
https://doi.org/10.3842/SIGMA.2022.030
Abstract. The combinatorial mutation of polygons, which transforms a given lattice
polygon into another one, is an important operation to understand mirror partners for
two-dimensional Fano manifolds, and the mutation-equivalent polygons give Q-Gorenstein
deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bi-
partite graph described on the real two-torus, one can assign a lattice polygon called the
perfect matching polygon. It is known that for each lattice polygon P there exists a dimer
model having P as the perfect matching polygon and satisfying certain consistency condi-
tions. Moreover, a dimer model has rich information regarding toric geometry associated
with the perfect matching polygon. In this paper, we introduce a set of operations which
we call deformations of consistent dimer models, and show that the deformations of consis-
tent dimer models realize the combinatorial mutations of the associated perfect matching
polygons.
Key words: dimer models; combinatorial mutation of polygons; mirror symmetry
2020 Mathematics Subject Classification: 52B20; 14M25; 14J33
Contents
1 Introduction 2
1.1 Background and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Dimer models and perfect matching polygons 4
2.1 What is a dimer model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Perfect matchings and the perfect matching polygon . . . . . . . . . . . . . . . . 5
3 Zigzag paths and their properties 8
3.1 Consistency conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Relationships between perfect matchings and zigzag paths . . . . . . . . . . . . . 10
4 Deformations of dimer models 14
4.1 Definition of deformations of dimer models . . . . . . . . . . . . . . . . . . . . . 14
4.2 Examples of deformations of dimer models . . . . . . . . . . . . . . . . . . . . . . 18
5 Combinatorial mutations of perfect matching polygons 19
5.1 Preliminaries on combinatorial mutations of polytopes . . . . . . . . . . . . . . . 19
5.2 The perfect matching polygons of deformed dimer models . . . . . . . . . . . . . 23
mailto:higashitani@ist.osaka-u.ac.jp
mailto:ynakaji@cc.kyoto-su.ac.jp
https://doi.org/10.3842/SIGMA.2022.030
2 A. Higashitani and Y. Nakajima
6 Extended deformations of dimer models 25
7 Foundations of extended deformations of dimer models 30
7.1 The proof of the non-degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.2 Behaviors of zigzag paths after extended deformations . . . . . . . . . . . . . . . 33
7.3 Properties of zigzag paths on deformed dimer models . . . . . . . . . . . . . . . . 37
7.4 Remarks on the extended deformations of hexagonal and rectangular dimer models 40
8 Combinatorial mutations of the PM polygon are realized
by extended deformations 42
A Mutations of dimer models 46
B Large examples 48
References 51
1 Introduction
1.1 Background and motivations
Fano manifolds are one of the well studied classes in geometry, and the classification of Fano
manifolds, which has been done in low dimensions, is a fundamental problem. Here, a Fano
manifold X is a complex projective manifold such that the anticanonical line bundle −KX is
ample. Recently, a new approach that uses mirror symmetry has been proposed for classifying
Fano manifolds as follows. First, a Fano manifold is expected to correspond to a certain Laurent
polynomial via mirror symmetry (see [8]). That is, a Laurent polynomial f ∈ C
[
x±1 , . . . , x
±
n
]
is
said to be a mirror partner for an n-dimensional Fano manifold X if the Taylor expansion of the
classical period πf of f coincides with a generating function for Gromov–Witten invariants of X
(see the references quoted above for the details of these terminologies). Furthermore, if a Fano
manifold X is a mirror partner of f , it is expected that X admits a toric degeneration XP . Here,
P := Newt(f) is the Newton polytope of f , which is defined as the convex hull of exponents
of monomials of f , and XP is the toric variety defined by the spanning fan of P (i.e., the
fan whose cones are spanned by the faces of P ). Thus, Laurent polynomials having the same
classical period are considered as mirror partners for the same Fano manifold X, and in general
there are many Laurent polynomials that are mirror partners for X. In order to understand the
relationship between such Laurent polynomials, an operation called the mutation of f , which
is a birational transformation analogue to a cluster transformation, was introduced in [12].
In particular, it was shown that if f, g ∈ C
[
x±1 , . . . , x
±
n
]
are transformed into each other by
mutations, then their classical periods are the same, that is, πf = πg [2, Lemma 1]. Moreover,
this mutation of Laurent polynomials induces an operation, called the combinatorial mutation,
which connects the associated Newton polytopes P = Newt(f) and Q = Newt(g) as defined in [2]
(see also Section 5.1). Also, it was shown in [18] that if P and Q are Fano polytopes and are
transformed into each other by combinatorial mutations, then the associated toric varieties XP
and XQ are related by a Q-Gorenstein (= qG) deformation; that is, there exists a flat family
X → P1 such that the relative canonical divisor is Q-Cartier and X0
∼= XP , X∞ ∼= XQ, where Xp
is the fiber of p ∈ P1. Thus, it has been conjectured that there is a bijection between qG-
deformation equivalence classes of “class TG” Fano manifolds and mutation equivalence classes
of Fano polytopes. There have previously been several affirmative results (e.g., [1, 22]).
Deformations of Dimer Models 3
1.2 Results
As mentioned above, the combinatorial mutation of polytopes is quite important in mirror sym-
metry of Fano manifolds. In this paper, we focus on the two-dimensional case, and the polygons
which we are interested in are not necessarily Fano. First, it is known that any lattice polygon
in R2 can be realized as the perfect matching polygon ∆Γ of a dimer model Γ satisfying the
consistency condition (see Sections 2 and 3). A dimer model is a bipartite graph on the real
two-torus (see Section 2 for more details), which was first introduced in the field of statistical
mechanics. Since the 2000s, string theorists have been using dimer models to study quiver gauge
theories (e.g., [11, 16, 23, 24] and references therein). As a result, relationships between dimer
models and many branches of mathematics have been discovered, e.g., see [6] and references
therein. (Note that a bipartite graph described on other surfaces, which is also called a dimer
model, has also been studied actively. For example, dimer models on the disk are related to
Grassmannians and their mirror symmetry, e.g., see [17, 29].) Based on this background, we
expect that there is a certain operation on a consistent dimer model that realizes the combinato-
rial mutation of the associated perfect matching polygon. In this paper, we introduce a concept
called the deformations of consistent dimer models. Note that there is another operation on
dimer models, called mutation of dimer models (see Appendix A). Although a mutation changes
the shape of a dimer model, it does not change the associated perfect matching polygon. Thus,
we need a new operation different from the mutation to realize our expectation.
To explain our main theorem, we briefly recall the combinatorial mutations of polygons
(see Section 5.1 for a more precise definition). Let N ∼= Z2 be a rank two lattice and M :=
HomZ(N,Z) ∼= Z2. First, we consider a lattice polygon P in NR := N⊗ZR and choose an edge E
of P . We then take a primitive inner normal vector w ∈ M for E, and consider the linear map
⟨w,−⟩ : NR → R. Using these notions, we determine the height ⟨w, u⟩ of each point u ∈ P . In
particular, a primitive lattice element uE ∈ N satisfying ⟨w, uE⟩ = 0 plays an important role
in defining the combinatorial mutation. Such an element uE is determined uniquely up to sign;
thus we fix one of them. Then, we define the line segment F := conv{0, uE}, which is called
a factor of P with respect to w. Using these data, we have the lattice polygon mutw(P, F ), which
is called the combinatorial mutation of P given by the vector w and the factor F , as defined
in Definition 5.2. In addition, we also define another combinatorial mutation mutw(P,−F ) in
a similar way. We note that although mutw(P, F ) looks different from mutw(P,−F ), they are
transformed into each other by a GL(2,Z)-transformation. On the other hand, for the given
lattice polygon P there exists a consistent dimer model Γ such that P = ∆Γ.
Based on this background, the deformation of a consistent dimer model is compatible with
the above combinatorial mutations in the following sense. Let Γ be the consistent dimer model.
The deformations of Γ are defined for a certain set of “zigzag paths” {z1, . . . , zr} on Γ corre-
sponding to the vector −w (see Section 3 concerning zigzag paths), and there are two kinds
of deformations which we call the zig-deformation and the zag-deformation, which are respec-
tively denoted by νzigp (Γ, {z1, . . . , zr}) and νzagp (Γ, {z1, . . . , zr}); see Definitions 4.3 and 4.4. For
some cases, these deformations are compatible with the combinatorial mutations of the perfect
matchings as follows.
Theorem 1.1 (see Proposition 4.5 and Theorem 5.10 for more details). In addition to the above
settings, we also assume that either one of the following conditions is satisfied:
(i) r = 1 or
(ii) Γ is a hexagonal or rectangular dimer model (see Definition 2.1).
Then the deformations νzigp (Γ, {z1, . . . , zr}) and νzagp (Γ, {z1, . . . , zr}) are consistent dimer models,
and we have
mutw(∆Γ, F ) = ∆
νzigp (Γ,{z1,...,zr}), mutw(∆Γ,−F ) = ∆νzagp (Γ,{z1,...,zr}),
4 A. Higashitani and Y. Nakajima
where ∆
νzigp (Γ,{z1,...,zr}) and ∆νzagp (Γ,{z1,...,zr}) are the perfect matching polygons of νzigp (Γ, {z1, . . . ,
zr}) and νzagp (Γ, {z1, . . . , zr}), respectively.
Note that the combinatorial mutations mutw(∆Γ,±F ) are defined for the perfect matching
polygon ∆Γ, whereas the deformations ν
zig/zag
p (Γ, {z1, . . . , zr}) are defined independently of ∆Γ.
When we drop the both conditions (i) and (ii) in Theorem 1.1, the deformed dimer model does
not necessarily realize the combinatorial mutation. In order to obtain the same results for a gen-
eral situation, we have to add some algorithmic operations to the definitions of deformations.
We formulate such operations as the extended zig-deformation νzigX (Γ, {z1, . . . , zr}) and extended
zag-deformation νzagY (Γ, {z1, . . . , zr}) as in Definitions 6.2 and 6.3. Then we can prove the state-
ment in Theorem 1.1 for the extended deformations (see Proposition 6.7 and Theorem 8.3).
By these results, the perfect matching polygons of the (extended) deformation of dimer models
satisfy the properties which are exactly the same as the combinatorial mutation of a polygon
(see Section 8).
Since dimer models are related with many branches of mathematics and physics, we con-
sidered that it would be of interest to compare certain objects (e.g., perfect matchings, the
associated toric variety) related with a consistent dimer model Γ to those of νzigX (Γ, {z1, . . . , zr})
(or νzagY (Γ, {z1, . . . , zr})).
The structure of this paper is as follows. In Section 2, we introduce dimer models and
related concepts. In particular, the notion of the perfect matching polygon introduced in this
section is one of main concepts in this paper. In Section 3, we introduce the notion of zigzag
paths, which are special paths on a dimer model. We then define the consistency condition
using zigzag paths, and discuss the relationships between perfect matchings and zigzag paths on
a consistent dimer model. Next, we focus specifically on type I zigzag paths, which are zigzag
paths having typical properties. Using these type I zigzag paths, we introduce the deformations
of consistent dimer models in Section 4. In Section 5, we compare the perfect matching polygon
of the deformed dimer model with the combinatorial mutation of the perfect matching polygon
of the original dimer model. In particular, we see that these polygons coincide in some cases
as in Theorem 1.1 (see Theorem 5.10). In Section 6, we consider some algorithmic operations
and define the extended deformations of consistent dimer models. In Section 7, we show the
fundamental properties of the extended deformation of consistent dimer models. In particular,
we prove the compatibility of the extended deformations and the combinatorial mutations for
general situations as in Theorem 8.3, and this induces Theorem 5.10. As we will mention in
Remark 6.5, the extended deformation of a consistent dimer model depends on the choice of
some of the data used in the processes (zig-5), (zag-5), as explained in Definitions 6.2 and 6.3
(see also Operation 6.6), whereas the perfect matching polygon of the extended deformation
of a consistent dimer model is determined uniquely. This ambiguity is caused by the fact that
there are several consistent dimer models giving the same perfect matching polygon. However,
it has been conjectured that such consistent dimer models are transformed into each other by
mutations of dimer models, and hence “conjecturally” our deformation of a consistent dimer
model is determined uniquely up to the mutation. We include a survey of this mutation of
dimer models in Appendix A. In Appendix B, we provide an additional example of the extended
deformation, the explanation of which is too lengthy to include in the main body of this paper.
2 Dimer models and perfect matching polygons
2.1 What is a dimer model?
We first introduce dimer models. Some ideas and concepts contained in this subsection are
originally derived from theoretical physics (e.g., [11, 16]).
Deformations of Dimer Models 5
A dimer model (or brane tiling) Γ is a finite bipartite graph on the real two-torus T := R2/Z2;
that is, the set Γ0 of nodes is divided into two parts Γ+
0 , Γ
−
0 , and the set Γ1 of edges consists
of the edges connecting nodes in Γ+
0 with those in Γ−
0 . In order to make the situation clear, we
color the nodes in Γ+
0 white, and those in Γ−
0 black. A connected component of T\Γ1 is called
a face of Γ, and we denote by Γ2 the set of faces. We also obtain the bipartite graph Γ̃ on R2
induced via the universal cover R2 → T. We call Γ̃ the universal cover of the dimer model Γ.
For example, the bipartite graph shown on the left side of Figure 2.1 is a dimer model where
the outer frame is the fundamental domain of T.
As the dual of a dimer model Γ, we define the quiver QΓ associated with Γ. Namely, we
assign a vertex dual to each face in Γ2, and an arrow dual to each edge in Γ1. The orientation
of arrows is determined so that the white node is on the right of the arrow. For example, the
right side of Figure 2.1 is the quiver associated with the dimer model on the left. Sometimes we
simply denote the quiver QΓ by Q.
1
0
3
2
Figure 2.1. A dimer model and the associated quiver.
The valency of a node is the number of edges incident to that node. We say that a node
on a dimer model is n-valent if its valency is n. We then define several operations on a dimer
model. The join move is the operation removing a 2-valent node and joining the two distinct
nodes connected to it as shown in Figure 2.2. Thus, using join moves we obtain a dimer model
having no 2-valent nodes. We say that a dimer model is reduced if it has no 2-valent nodes.
We can see that the quiver associated with a reduced dimer model contains no 2-cycles. On
the other hand, there is the operation called the split move, which inserts a 2-valent node (see
Figure 2.2).
We say that two reduced dimer models Γ, Γ′ are isomorphic, which is denoted by Γ ∼= Γ′, if
their underlying cell decompositions of T are homotopy-equivalent.
join move
split move
Figure 2.2. An example of the join and split move.
We sometimes focus on the following type of dimer models.
Definition 2.1. We say that a dimer model Γ is hexagonal (resp. rectangular) if any face
of Γ is hexagon (resp. rectangle) and any node of Γ is 3-valent (resp. 4-valent). In particular,
a hexagonal (resp. rectangular) dimer model is homotopy-equivalent to a dimer model whose
faces are all regular hexagons (resp. squares).
2.2 Perfect matchings and the perfect matching polygon
Next, we assign a lattice polygon to each dimer model. For this purpose, we will introduce the
notion of perfect matchings, and we construct a polygon called the perfect matching polygon.
6 A. Higashitani and Y. Nakajima
Definition 2.2. A perfect matching (or dimer configuration) on a dimer model Γ is a subset P
of Γ1 such that each node is the end point of precisely one edge in P.
In general, not every dimer model necessarily has a perfect matching. In this paper, we
will mainly discuss consistent dimer models, and such dimer models have perfect matchings.
Moreover, we can extend a perfect matching P to one on Γ̃ via the universal cover R2 → T.
We call this a perfect matching on Γ̃, and use the same notation P. For example, some perfect
matchings on the dimer model given in Figure 2.1 are shown in Figure 2.3. (This dimer model
has eight perfect matchings in total.)
P0 P1 P2 P3 P4
Figure 2.3. Some perfect matchings on the dimer model given in Figure 2.1.
We say that a dimer model is non-degenerate if every edge is contained in some perfect
matchings. It is known that this non-degeneracy condition is equivalent to the strong marriage
condition; that is, the dimer model has equal numbers of black and white nodes and every
proper subset of the black nodes of size n is connected to at least n + 1 white nodes (e.g., [7,
Remark 2.12]).
Following [20, Section 5], we next define the perfect matching polygon. We first fix a perfect
matching P0, and call this the reference perfect matching. For any perfect matching P, we
consider the connected components of the universal cover R2 divided by P ∪ P0. Then, we
consider the height function hP,P0 , which is a locally constant function on R2\(P ∪ P0), defined
as follows. First, we choose a connected component of R2\(P∪P0), and define the value of hP,P0
as 0. Then, this function increases by 1 when we cross
– an edge e ∈ P with the black node on the right, or
– an edge e ∈ P0 with the white node on the right,
and decreases by 1 when we cross
– an edge e ∈ P with the white node on the right, or
– an edge e ∈ P0 with the black node on the right.
This function is determined up to a choice of a connected component of value 0. For example,
Figure 2.4 shows the height function hP2,P3 on the dimer model given in Figure 2.1, where the
red square stands for a fundamental domain of T, the edges in P2 (resp. P3) are colored blue
(resp. green), and the number filled in each component is the value of hP2,P3 .
We then take a point pt ∈ R2\(P ∪ P0), and define the height change
h(P,P0) = (hx(P,P0), hy(P,P0)) ∈ Z2
of P with respect to P0 as the differences of the height function:
hx(P,P0) = hP,P0(pt+ (1, 0))− hP,P0(pt),
hy(P,P0) = hP,P0(pt+ (0, 1))− hP,P0(pt).
Deformations of Dimer Models 7
-1
-1
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
Figure 2.4. The height function hP2,P3 .
We remark that this does not depend on the choice of pt ∈ R2\(P ∪ P0). We then consider the
height change h(P,P′) for any pair of perfect matchings P, P′, but since we have
h(P,P′) = h(P,P0)− h(P′,P0),
it is enough to consider height changes with respect to the reference perfect matching P0. Then,
the perfect matching (= PM ) polygon (or characteristic polygon) ∆Γ ⊂ R2 of a dimer model Γ
is defined as the convex hull of
{
h(P,P0) ∈ Z2 |P ∈ PM(Γ)
}
where PM(Γ) is the set of perfect
matchings on Γ.
Remark 2.3. The description of height changes depends on the choice of the coordinate system
fixed in T (i.e., the choice of a fundamental domain). A change of a coordinate system induces
a GL(2,Z)-action on the PM polygon, and this action does not affect our problem. In the
following, we say that two polygons P and Q are GL(2,Z)-equivalent if they are transformed
into each other by GL(2,Z)-transformations, in which case we denote P ∼= Q. Thus, we may
fix some fundamental domain of T. Also, we remark that the description of the polygon ∆Γ
depends on the choice of a reference perfect matching, but it is determined up to translations.
Definition 2.4. Fix a perfect matching P0 and let ∆Γ be the perfect matching polygon. We
say that a perfect matching P is
� a corner (or extremal) perfect matching if h(P,P0) is a vertex of ∆Γ,
� a boundary (or external) perfect matching if h(P,P0) is a lattice point on an edge of ∆Γ
(in particular, corner perfect matchings are boundary perfect matchings),
� an internal perfect matching if h(P,P0) is an interior lattice point of ∆Γ.
In the next subsection, we will introduce consistent dimer models (see Definition 3.2), which
have several nice properties. If a dimer model is consistent, then there exists a unique corner
perfect matching corresponding to each vertex of ∆Γ (e.g., [7, Corollary 4.27], [20, Proposi-
tion 9.2]). Thus, we can give a cyclic order to corner perfect matchings along the corresponding
vertices of ∆Γ in the anti-clockwise direction. We say that two corner perfect matchings are
adjacent if they are adjacent with respect to the given cyclic order.
Example 2.5. We consider the dimer model given in Figure 2.1, which is consistent as we will
see in the next subsection. Fix the perfect matching P0 shown in Figure 2.3 as the reference
one. Then, we see that the perfect matchings P1, . . . ,P4 correspond to lattice points (1, 0), (1, 1),
(−1, 0), (0,−1), respectively. Since P0 is the reference perfect matching, it corresponds to (0, 0).
In addition, this dimer model has three perfect matchings that are not listed in Figure 2.3, and
such perfect matchings also correspond to (0, 0). Thus, the PM polygon takes the form shown
in Figure 2.5, and hence P1, . . . ,P4 are corner perfect matchings.
8 A. Higashitani and Y. Nakajima
P1
P2
P3
P4
Figure 2.5. The PM polygon of the dimer model given in Figure 2.1.
In this way, we can obtain the PM polygon from a dimer model. On the other hand, it is
known that any lattice polygon can be obtained as the PM polygon of a certain dimer model.
Theorem 2.6 ([14, 20]). For any lattice polygon ∆ in R2, there exists a dimer model Γ giving ∆
as the PM polygon ∆Γ. Furthermore, we can take this Γ as it satisfies the consistency condition
(see Definition 3.2).
Thus, for a given lattice polygon ∆, we say that Γ is a dimer model associated with ∆ if
the PM polygon of Γ coincides with ∆. We remark that for a given polygon ∆, the associated
consistent dimer model is not unique in general.
3 Zigzag paths and their properties
3.1 Consistency conditions
In this subsection, we introduce the consistency condition. In order to define this condition,
we first introduce the notion of zigzag paths. These paths are also the main ingredients for
introducing deformations of dimer models.
Definition 3.1. We say that a path on a dimer model is a zigzag path if it makes a maximum
turn to the right on a white node and a maximum turn to the left on a black node. Also, we say
that a zigzag path is reduced if it does not pass through 2-valent nodes. (We remark that we can
make a zigzag path reduced using the join moves. In particular, any zigzag path on a reduced
dimer model is reduced.)
Since a dimer model has only finitely many edges, we see that all zigzag paths are periodic.
For a zigzag path z on Γ, we define the length of z, which is denoted by ℓ(z), as the number
of edges of Γ constituting z. In particular, we see that ℓ(z) is an even integer. Thus, edges on
a zigzag path are indexed by elements in Z/(2n)Z for some integer ℓ(z)/2 = n ≥ 1. Fix a black
node on a zigzag path z as the starting point of z, and denote z as a sequence of edges starting
from the fixed black node: z = z[1]z[2] · · · z[2n− 1]z[2n].
z[1]
z[2]
z[3]
z[4]
z[5]
z[6]z[2n]
An edge in a zigzag path z is called a zig (resp. zag) of z if it is indexed by an odd (resp. even)
integer. We denote by Zig(z) (resp. Zag(z)) the set of zigs (resp. zags) appearing in a zigzag
path z, which is a finite set. Two zigzag paths are said to intersect if they share an edge (not
a node). We note that if z does not have a self-intersection, Zig(z) and Zag(z) are disjoint sets.
For any edge e of a dimer model, we can consider the zigzag path containing e as a zig and the
Deformations of Dimer Models 9
zigzag path containing e as a zag. Thus, any edge e is contained in at most two zigzag paths. If
such zigzag paths do not have a self-intersection, e is contained in exactly two zigzag paths. For
example, zigzag paths on the dimer model given in the left of Figure 2.1 are shown in Figure 3.1.
z1 z2 z3 z4
Figure 3.1. All zigzag paths on the dimer model given in Figure 2.1.
For a zigzag path z on a dimer model Γ, we also consider the lift of z to the universal cover Γ̃.
Let z̃(α) denote a zigzag path on Γ̃ whose projection on Γ is z where α ∈ Z. When we do not
need to specify these, we simply denote each of them by z̃. Then, we see that a zigzag path on Γ̃
is either periodic or infinite in both directions. Using these notions, we introduce the consistency
condition.
Definition 3.2 (see [19, Definition 3.5]). We say that a dimer model is (zigzag) consistent if it
satisfies the following conditions:
(1) there is no homologically trivial zigzag path,
(2) no zigzag path on the universal cover has a self-intersection,
(3) no pair of zigzag paths on the universal cover intersect each other in the same direction
more than once. That is, if a pair of zigzag paths (z̃, w̃) on the universal cover has two
intersected edges a1, a2 and z̃ points from a1 to a2, then w̃ points from a2 to a1.
In the literature, there are several conditions that are equivalent to Definition 3.2 (for more
details, see [4, 19]), and it is known that a consistent dimer model is non-degenerate (e.g., [20,
Proposition 8.1]). For example, we see that the dimer model given in Figure 2.1 is consistent by
checking all zigzag paths, which are shown in Figure 3.1. We also remark that this dimer model
satisfies the stronger condition called isoradial (see Definition 3.4).
In this paper, we also use another condition called properly ordered. To explain the proper
ordering, we prepare several notation. First, considering a zigzag path z as a 1-cycle on T, we
have the homology class [z] ∈ H1(T) ∼= Z2. We call this element [z] ∈ Z2 the slope of z. We
remark that even if we apply the join and split moves to nodes contained in a zigzag path, such
operations do not change the slope. If a zigzag path does not have any self-intersection, the slope
of each zigzag path is primitive. Now, we consider slopes (a, b) ∈ Z2 of zigzag paths that are
not homologically trivial. The set of such slopes has a natural cyclic order by considering (a, b)
as an element of the unit circle:
(a, b)√
a2 + b2
∈ S1.
We say that two zigzag paths are adjacent if their slopes are adjacent with respect to the above
cyclic order. Using this cyclic order, we define a properly ordered dimer model below. In
particular, it is known that a dimer model is consistent in the sense of Definition 3.2 if and only
if it is properly ordered (see [19, Proposition 4.4]).
Definition 3.3 (see [14, Section 3.1]). A dimer model is said to be properly ordered if
10 A. Higashitani and Y. Nakajima
(1) there is no homologically trivial zigzag path,
(2) no zigzag path on the universal cover has a self-intersection,
(3) no pair of zigzag paths with the same slope have a common node,
(4) for any node on the dimer model, the natural cyclic order on the set of zigzag paths
incident to that node coincides with the cyclic order determined by their slopes.
We also introduce isoradial dimer models which are stronger than consistent ones. The dimer
model given in Figure 2.1 is isoradial in particular.
Definition 3.4 ([25, Theorem 5.1]; see also [10, 26]). We say that a dimer model Γ is isoradial
(or geometrically consistent) if
(1) every zigzag path is a simple closed curve,
(2) any pair of zigzag paths on the universal cover share at most one edge.
3.2 Relationships between perfect matchings and zigzag paths
We can now discuss the relationship between perfect matchings and zigzag paths. The following
proposition is essential throughout this paper.
Proposition 3.5 (see [14, Theorem 3.3 and Corollary 3.8], [20, Proposition 9.2 and Corol-
lary 9.3]). There exists a one-to-one correspondence between the set of slopes of zigzag paths on
a consistent dimer model Γ and the set of primitive side segments of the PM polygon ∆Γ. More
precisely, each slope of a zigzag path is the primitive outer normal vector for a primitive side
segment of ∆Γ.
Moreover, zigzag paths having the same slope arise as the difference of two adjacent corner
perfect matchings P, P′ (i.e., the edges in P ∪ P′\P ∩ P′ form zigzag paths). Thus, any corner
perfect matching intersects with half of the edges constituting a certain zigzag path.
For example, the zigzag path z1 shown in Figure 3.1 is obtained from the pair of adjacent
corner perfect matchings (P1,P2) given in Figure 2.3. Also, the zigzag paths z2, z3, z4 are
obtained by pairs (P2,P3), (P3,P4), (P4,P1), respectively.
By Proposition 3.5, we can assign each edge of the PM polygon to a zigzag path z; thus we
will call this the edge corresponding to z. In particular, the edges corresponding to zigzag paths
having the same slope are all the same.
Let P, P′ be adjacent corner perfect matchings on a consistent dimer model, and z1, . . . , zr be
the zigzag paths arising from P and P′ as in Proposition 3.5. In particular, these zigzag paths
have the same slope. We see that P∩ zi = Zig(zi) and P′ ∩ zi = Zag(zi) (or P∩ zi = Zag(zi) and
P′ ∩ zi = Zig(zi)) for any i = 1, . . . , r. Here, P ∩ zi denotes the subset of edges in P contained
in zi. Then, we have the description of boundary perfect matchings using the corner ones.
Proposition 3.6 (e.g., [7, Proposition 4.35], [14, Corollary 3.8]). Let P, P′ and z1, . . . , zr be as
above. Let E be the edge of the PM polygon of Γ corresponding to z1, . . . , zr. We assume that
P ∩ zi = Zig(zi) and P′ ∩ zi = Zag(zi). Then, any boundary perfect matching corresponding to
a lattice point on E can be described as(
P\
⋃
i∈I
Zig(zi)
)
∪
⋃
i∈I
Zag(zi) or
(
P′\
⋃
i∈I
Zag(zi)
)
∪
⋃
i∈I
Zig(zi),
where I is a subset of {1, . . . , r}. In particular, the number of perfect matchings corresponding to
a lattice point q on E is
(
r
m
)
, where m is the number of primitive side segments of E between q
and one of the endpoints of E.
Deformations of Dimer Models 11
We then observe the relationships between zigzag paths and height changes of perfect match-
ings. Some of these relationships are well-known to experts, but we will consider them in detail
because these statements are quite important when we define the deformation of consistent
dimer models in Section 4, and also for the self-containedness.
Observation 3.7 (cf. [20, Section 5.3]). Let Γ be a consistent dimer model. For any zigzag
path z, the slope [z] is an element in H1(T). On the other hand, we can consider height changes as
elements in the cohomology group H1(T) ∼= Z2. We consider a pairing ⟨−,−⟩ : H1(T)×H1(T)→Z.
By Propositions 3.5 and 3.6, there is a perfect matching P′ that intersects half of the edges
constituting z. Then, for any perfect matching P, we have ⟨h(P,P′), [z]⟩ ≤ 0. In fact, we first
replace z by the path pz on the quiver QΓ going along the left side of z (see the figure below).
z
pz
Then, considering this path pz as the element [pz] ∈ H1(T), we have [z] = [pz]. By a choice
of P′, this pz does not cross any edge in P′, and if pz crosses an edge in P, we can see the white
node on the right by the definition of QΓ. Thus, we have the desired inequality.
For a perfect matching P and a zigzag path z on a dimer model Γ, we denote by |P ∩ z|
the number of edges in P ∩ z. Since the number of perfect matchings is finite, the maximum
(resp. minimum) number ωmax(z) (resp. ωmin(z)) of |P ∩ z| exists for each zigzag path z. For
a consistent dimer model, z can be obtained as the difference of adjacent perfect matchings (see
Proposition 3.6); thus we clearly have ℓ(z)/2 = ωmax(z). We set
PMmax(z) = {P ∈ PM(Γ) | |P ∩ z| = ωmax(z)},
PMmin(z) = {P ∈ PM(Γ) | |P ∩ z| = ωmin(z)}.
In particular, if P, P′ are adjacent corner perfect matchings on a consistent dimer model Γ, and z
is one of the zigzag paths obtained by P, P′, then we have P∩z = Zig(z) and P′∩z = Zag(z) (or
P∩z = Zag(z) and P′∩z = Zig(z)), and hence the next lemma easily follows from Propositions 3.5
and 3.6.
Lemma 3.8. Let z be a zigzag path on a consistent dimer model Γ, and E be the edge of the
PM polygon of Γ corresponding to z. If P1, . . . ,Ps are boundary perfect matchings corresponding
to lattice points on E, then we have {P1, . . . ,Ps} = PMmax(z), and ωmax(z) = |Pi ∩ z| = ℓ(z)/2
for any i = 1, . . . , s.
Next, we prepare several lemmas, which play crucial roles to define deformations of consistent
dimer models.
Lemma 3.9. Let the notation be the same as Lemma 3.8. For any perfect matching P, we have
|P ∩ z| = ℓ(z)/2− ⟨h(P,Pi),−[z]⟩.
In particular, we have
⟨h(P,Pi),−[z]⟩ ≤ ωmax(z)− ωmin(z),
and the equality holds for P ∈ PMmin(z).
12 A. Higashitani and Y. Nakajima
Proof. First, the maximum number of |P ∩ z| is ℓ(z)/2, in which case P = Pi by Lemma 3.8.
If the path pz as in Observation 3.7 crosses an edge e in P, it means that e is not an edge
constituting z, and thus any edge sharing the same white node as e is not contained in P. By
Observation 3.7, we see that for any perfect matching P the number of edges in P intersecting
with pz coincides with −⟨h(P,Pi), [z]⟩, and thus we have the first equation.
The second assertion follows from the first equation and Lemma 3.8. ■
By this lemma, we see that P ∈ PMmin(z) if and only if ⟨h(P,Pi), [z]⟩≤⟨h(P′,Pi), [z]⟩ for any
P′ ∈ PM(Γ). Thus, we see that P ∈ PMmin(z) lies either on a vertex of the PM polygon ∆Γ or
an edge of ∆Γ. Also, even if two zigzag paths zj and zk have the same slope, ℓ(zj) ̸= ℓ(zk) and
|P∩zj | ≠ |P∩zk| in general, but the difference of these values is the same in the following sense:
Lemma 3.10. Let Γ be a consistent dimer model, and z, z′ be zigzag paths on Γ having the
same slope. Then, for any perfect matching P, we have
ℓ(z)/2− |P ∩ z| = ℓ(z′)/2− |P ∩ z′|.
In particular, we have
ℓ(z)/2− ωmin(z) = ℓ(z′)/2− ωmin(z
′).
Proof. Since [z] = [z′], the first equation follows from Lemma 3.9. Considering a perfect
matching P such that the value of ⟨h(P,Pi),−[z]⟩ = ⟨h(P,Pi),−[z′]⟩ is maximal, we have the
second equation. ■
We then divide zigzag paths on a consistent dimer model into the following two types. Note
that type I zigzag paths are used to define the deformation of consistent dimer models.
Definition 3.11. Let Γ be a dimer model, and z be a zigzag path on Γ.
(1) We say that z is type I if z is reduced and z̃ intersects with any other zigzag path on the
universal cover Γ̃ at most once.
(2) We say that z is type II if z is reduced and there exists a zigzag path w̃ on the universal
cover Γ̃ such that w̃ intersects with z̃ in the opposite direction more than once.
We note that any zigzag path on a reduced consistent dimer model is either type I or type II.
In particular, if Γ is isoradial, then all zigzag paths are type I (see Definition 3.4).
As the following lemmas show, the properties of type I zigzag paths are particularly nice.
Lemma 3.12. Let z be a type I zigzag path on a consistent dimer model Γ. Then, there exists
a perfect matching P on Γ satisfying |P ∩ z| = 0, which means that P is in PMmin(z).
Proof. In order to find a perfect matching P, we will use the method discussed in [14, Section 3],
[7, Section 4]. For this, we first prepare some notation.
We consider the sequence [z1], . . . , [zn] of slopes of zigzag paths on Γ. Since Γ is consistent,
it is properly ordered, and thus we can assume that the slopes are ordered cyclically with this
order. We note that some of the slopes may coincide. Then, we define the normal fan in
H1(T) ⊗Z R whose rays are slopes [z1], . . . , [zn]. In particular, each two-dimensional cone σ is
generated by different adjacent slopes. We denote by θi the angle formed by [zi]. Here, we
suppose that z = zk. Let R be a ray whose angle is θk +π+ ϵ where ϵ > 0 is a sufficiently small
angle satisfying the condition that θk + π + ϵ does not coincide with any θi.
Then, for each node v ∈ Γ0, we define the fan ξ(v) generated by the slopes of zigzag paths
factoring through v. In this fan ξ(v), we can find the zigzag path z′v whose slope makes the
smallest clockwise angle with R, and the zigzag path z′′v whose slope makes the smallest anti-
clockwise angle with R. Since Γ is properly ordered, these zigzag paths are consecutive around v.
The intersection of z′v and z′′v is the edge e(v) which has v as an endpoint.
Deformations of Dimer Models 13
v v′
e(v) = e(v′)
z′vz′′v
We then apply the same argument to the node v′ which is the other endpoint of e(v). The
proper ordering on Γ leads to the conclusion that e(v) = e(v′) (see the above figure). We repeat
these arguments for any node, but it is enough to consider e(v)’s for any v ∈ Γ+
0 (or v ∈ Γ−
0 ).
By [14, Section 3.2] or [7, Lemma 4.19], we see that the subset of edges e(v) for all v ∈ Γ+
0 forms
a perfect matching. Furthermore, since z = zk is type I, there exists a zigzag path whose slope
is located at an angle less than π in an anti-clockwise (resp. clockwise) direction from [z] in ξ(v)
by [7, Lemma 4.11 and its proof], in which case such a slope is located between [z] and [z′v]
(resp. [z′′v ]) or coincides with [z′v] (resp. [z
′′
v ]). Thus, we have z ̸= z′v and z ̸= z′′v by the definition
of the ray R. Therefore, in this case e(v) is not contained in z by the above construction. Also,
we see that if a node v does not lie on z, e(v) is not contained in z. Thus, the perfect matching
constructed in the above fashion satisfies the desired condition. ■
Lemma 3.13. Let z be a type I zigzag path on a consistent dimer model. Then, we have
ωmin(z) = 0, and hence ℓ(z) is the same for all type I zigzag paths having the same slope.
Proof. This follows from Lemmas 3.10 and 3.12. ■
For zigzag paths z, w on a dimer model Γ, we denote by z ∩ w the subset of edges that are
intersections of z and w on Γ. We remark that if z is type I then the number of intersections
of z̃ and w̃ on Γ̃ is less than or equal to one, but there are possibly more intersections of z and w
if we consider them on Γ.
Lemma 3.14. Let z be a type I zigzag path on a reduced consistent dimer model Γ. We suppose
that a zigzag path w has intersections with z on Γ. Then, we see that z ∩ w ⊂ Zig(z) or
z ∩ w ⊂ Zag(z).
Proof. Let e1, e2 be edges of Γ, and we assume that w intersects with z at e1 and e2. If ei is
a zig of z, then it is a zag of w, and vice versa. We assume that e1 is a zig of z and e2 is a zag
of z.
We then lift these edges on the universal cover Γ̃. Let ẽ1, ẽ2 be edges of Γ̃ whose restrictions
on Γ are e1, e2, respectively. In particular, ẽ1 is a zig of z̃ and ẽ2 is a zag of z̃. Then, there exist
zigzag paths w̃1, w̃2 on Γ̃ whose restriction on Γ is just w, and w̃1 (resp. w̃2) intersects with z̃
at ẽ1 (resp. ẽ2). The zigzag path z̃ splits R2 into two pieces, and w̃1 (resp. w̃2) intersects with z̃
from left to right (resp. right to left) by the definition of zigzag paths (see the figure below).
z̃
w̃2
w̃1
right of z̃
left of z̃
Since z is type I, there is no intersections of z̃ and w̃i except ẽi where i = 1, 2. Thus, we can
not superimpose w̃1 and w̃2 using translations. This contradicts a choice of w̃1, w̃2. ■
14 A. Higashitani and Y. Nakajima
The following lemma follows from the argument in the proof of [7, Proposition 3.12].
Lemma 3.15. Let z, w be zigzag paths on a consistent dimer model Γ. We assume that z̃
intersects with w̃ on the universal cover Γ̃ at most once. Then, the slopes [z], [w] are linearly
independent if and only if z̃ and w̃ intersect on precisely one edge.
Lemma 3.16. Let z1, . . . , zr be type I zigzag paths on a reduced consistent dimer model Γ having
the same slope. We suppose that a zigzag path w has intersections with zj for some j. Then,
w intersects with every zi (i = 1, . . . , r), and the intersections w∩z1, . . . , w∩zr are all in Zig(w)
or all in Zag(w). Moreover, we have |w ∩ z1| = · · · = |w ∩ zr|.
Proof. Since z1, . . . , zr are type I, w̃ intersects with each z̃i at most once. Thus, each pair
of zigzag paths (w̃, z̃i) for i = 1, . . . , r satisfies the assumption in Lemma 3.15. Since w has
intersections with zj , w̃ intersects with z̃j precisely once on the universal cover. By Lemma 3.15,
we see that [w] and [zj ] are linearly independent, and hence [w] and [zi] are linearly independent
for any i. Thus, w̃ intersects with z̃i precisely once for any i = 1, . . . , r.
The latter assertion follows from a similar argument as in the proof of Lemma 3.14. More
precisely, since w̃ intersects with z̃i precisely once for any i = 1, . . . , r, if w̃ intersects with z̃i
from right to left (resp. left to right), then so do zigzag paths having the same slope. This means
that intersections are all in Zig(w) (resp. all in Zag(w)).
Also, let z̃1 and z̃1
′ be zigzag paths on Γ̃ that are projected onto the zigzag path z1 on Γ.
We assume that there is no zigzag path projected onto z1 between z̃1 and z̃1
′. Also, we assume
that w̃ first intersects with z̃1, then intersects with z̃1
′. Then, for all i = 2, . . . , r we can find
a unique zigzag path z̃i on Γ̃ such that it is projected onto zi and is located between z̃1 and z̃1
′.
Thus, after w̃ intersects with z̃1, it intersects with z̃2, . . . , z̃r precisely once and then arrives
at z̃1
′. We can apply the same arguments for any pair (z̃1, z̃1
′) of zigzag paths on Γ̃ satisfying
the above properties. Thus, projecting onto Γ we have |w ∩ z1| = · · · = |w ∩ zr|. ■
4 Deformations of dimer models
In this section, we will introduce the concept of a deformation of consistent dimer models. This
operation is defined for type I zigzag paths on a consistent dimer model, and there are two kinds
of deformations, which we call the zig-deformation (see Definition 4.3) and the zag-deformation
(see Definition 4.4). For some special cases, these deformations preserve the consistency con-
dition (see Proposition 4.5), but they change the associated PM polygon, whereas we will see
that the PM polygon of the deformed dimer model is exactly the combinatorial mutation of
a polygon (see Theorem 5.10).
4.1 Definition of deformations of dimer models
Let Γ be a reduced consistent dimer model. In particular, any slope of a zigzag path on Γ
is primitive. Let Zv(Γ) be the subset of zigzag paths on Γ whose slopes are the primitive
vector v ∈ Z2, and ZI
v(Γ) be the subset of Zv(Γ) consisting of type I zigzag paths. We first
prepare the deformation data.
Definition 4.1 (deformation data). Let Γ be a reduced consistent dimer model. In order to
define the deformation of Γ, we fix the following data.
(1) We choose a type I zigzag path z, and let 2n := ℓ(z) and v := [z].
(2) We then fix a positive integer r such that r ≤ |ZI
v(Γ)|, and let h = n− r.
Deformations of Dimer Models 15
(3) We take a subset {z1, . . . , zr} ⊂ ZI
v(Γ) of type I zigzag paths, in which case we have
2n = ℓ(z1) = · · · = ℓ(zr) by Lemma 3.13. Therefore, each zi can be described as
zi = zi[1]zi[2] · · · zi[2n− 1]zi[2n].
(4) We consider non-negative integers p1, . . . , pr ∈ Z≥0 such that p1 + · · · + pr = n − r = h.
We call p := (p1, . . . , pr) a deformation weight of {z1, . . . , zr}.
Remark 4.2. To define the deformation data, we need a type I zigzag path. If a dimer model
is isoradial, then any zigzag path is type I (see Definition 3.4), and hence
∣∣ZI
v(Γ)
∣∣ = |Zv(Γ)|.
Also, even if Γ contains no type I zigzag paths, we sometimes change a type II zigzag path into
type I by using the mutations of dimer models (see Appendix A, especially Example A.4).
Definition 4.3 (zig-deformation). Let the notation be the same as in Definition 4.1. For
a deformation weight p = (p1, . . . , pr) of {z1, . . . , zr}, we consider the following procedures:
(zig-1) Using split moves, we insert pi white nodes and pi black nodes in each zig of zi.
[Notation]
� For any zig zi[2m − 1] of zi where m = 1, . . . , n and i = 1, . . . , r, we denote by
bi[2m − 1] (resp. wi[2m − 1]) the black (resp. white) node that is the endpoint of
zi[2m− 1].
� We denote the white nodes added in the zig zi[2m−1] by wi,1[2m−1], . . . , wi,pi [2m−
1], and the black ones by bi,1[2m−1], . . . , bi,pi [2m−1]. Here, the subscripts increase
in the direction from bi[2m− 1] to wi[2m− 1].
(zig-2) We remove all zags of zi for every i = 1, . . . , r.
(zig-3) If pi ̸= 0, then we connect the white node wi,j [2m−1] to the black node bi,j [2m+1] where
j = 1, . . . , pi and m = 1, . . . , n. (Note that wi,j [2n − 1] is connected to bi,j [2n + 1] :=
bi,j [1].) We denote by zi,j the new 1-cycle, which will be a zigzag path on the deformed
dimer model, obtained by connecting
wi,j [2n− 1], bi,j [2n− 1], wi,j [2n− 3], bi,j [2ni − 3], . . . , wi,j [1], bi,j [1]
cyclically (i = 1, . . . , r and j = 1, . . . , pi).
(join) If there exist 2-valent nodes, then we apply the join moves to the dimer model obtained
by the above procedures and make it reduced.
We denote the resulting dimer model by νzigp (Γ, {z1, . . . , zr}), and call it the zig-deformation
of Γ at {z1, . . . , zr} with respect to the weight p. If a situation is clear, we simply denote this
by νzigp (Γ).
Similarly, we can define the “zag version” of this deformation as follows.
Definition 4.4 (zag-deformation). Let the notation be the same as Definition 4.1. For a defor-
mation weight p = (p1, . . . , pr) of {z1, . . . , zr}, we consider the following procedures:
(zag-1) Using split moves, we insert pi white nodes and pi black nodes in each zag of zi.
[Notation]
� For any zag zi[2m] of zi where m = 1, . . . , n and i = 1, . . . , r, we denote by wi[2m]
(resp. bi[2m]) the white (resp. black) node that is the endpoint of zi[2m].
� We denote the white nodes added in the zag zi[2m] by wi,1[2m], . . . , wi,pi [2m],
and the black ones by bi,1[2m], . . . , bi,pi [2m]. Here, the subscripts increase in the
direction from wi[2m] to bi[2m].
16 A. Higashitani and Y. Nakajima
zi[2m+ 1]
zi[2m]
zi[2m− 1]
(zig-1)
(zig-2) (zig-3)
Figure 4.1. The zig-deformation at zi with pi = 2.
(zag-2) We remove all zigs of zi for every i = 1, . . . , r.
(zag-3) If pi ̸= 0, then we connect the black node bi,j [2m] to the white node wi,j [2m+ 2] where
j = 1, . . . , pi and m = 1, . . . , n. (Note that bi,j [2n] is connected to wi,j [2n+2] := wi,j [2].)
We denote by zi,j the new 1-cycle, which will be a zigzag path on the deformed dimer
model, obtained by connecting
bi,j [2n], wi,j [2n], bi,j [2n− 2], wi,j [2n− 2], . . . , bi,j [2], wi,j [2]
cyclically (i = 1, . . . , r and j = 1, . . . , pi).
(join) If there exist 2-valent nodes, then we apply the join moves to the dimer model obtained
by the above procedures and make it reduced.
We denote the resulting dimer model by νzagp (Γ, {z1, . . . , zr}), and call it the zag-deformation
of Γ at {z1, . . . , zr} with respect to the weight p. If a situation is clear, we simply denote this
by νzagp (Γ).
For some special cases, these deformations preserve the consistency condition as in Proposi-
tion 4.5 below. Note that in Section 6 we will introduce extended deformations of dimer models
which always preserve the consistency condition (see Proposition 6.7).
Proposition 4.5. Let the notation be the same as Definitions 4.1, 4.3 and 4.4. We assume that
either one of the following conditions is satisfied:
(i) r = 1 or
(ii) Γ is a hexagonal or rectangular dimer model (see Definition 2.1).
Then, the dimer models νzigp (Γ, {z1, . . . , zr}) and νzagp (Γ, {z1, . . . , zr}) are consistent.
Proof. This follows from Proposition 6.7 (see also Remark 6.4 and Proposition 7.12). ■
We note that for some cases the zig-deformation and zag-deformation are mutually inverse
operations on the level of dimer models as in Proposition 4.6 below. To observe this, let
νzigp (Γ, {z1, . . . , zr}) be the zig-deformed dimer model as in Definition 4.3. We assume that
pi = 1 for some i and consider the zigzag path zi,pi = zi,1 on νzigp (Γ) created in place of zi on Γ
Deformations of Dimer Models 17
zi[2m+ 2]
zi[2m+ 1]
zi[2m]
(zag-1)
(zag-2) (zag-3)
Figure 4.2. The zag-deformation at zi with pi = 2.
via the zig-deformation. Note that zi,1 is type I (see Proposition 7.5). Then, we take a new
deformation data q so that zi,1 is chosen in the data with the weight 1, and consider the zag-
deformation of νzigp (Γ) at some set of zigzag paths with respected to q. Through this operation,
zi,1 changes into the original zigzag path zi as in Figure 4.3.
zi zi,1
νzigp νzagq
Figure 4.3. Transitions of zi via the composition of the zig-deformation and zag-deformation. (In
the rightmost figure, we still do not apply (join). The zigzag path in this figure coincides with zi after
applying (join).)
The next proposition follows from these observations.
Proposition 4.6. We consider the situation as in Definition 4.1, and assume that p1 = · · · =
pr = 1, in which case r = h. Let zi,pi = zi,1 (i = 1, . . . , r) be the zigzag path of νzigp (Γ, {z1, . . . , zr})
(resp. νzagp (Γ, {z1, . . . , zr})) created by modifying zi as in Definition 4.3 (resp. Definition 4.4).
We take q = (1, . . . , 1) as a deformation weight of {z1,1 , . . . , zr,1}. Then, we respectively have
νzagq
(
νzigp (Γ, {z1, . . . , zr}), {z1,1 , . . . , zr,1}
)
= Γ,
νzigq
(
νzagp (Γ, {z1, . . . , zr}), {z1,1 , . . . , zr,1}
)
= Γ.
This property depends on a choice of a deformation data, thus in general these deformations
are not mutually inverse as in Example 4.9 given in the next subsection. Whereas, on the level
of the PM polygons, they are mutually inverse as we will see in Corollary 8.6.
18 A. Higashitani and Y. Nakajima
4.2 Examples of deformations of dimer models
In this subsection we give several examples for the case of r = 1, in which the deformed dimer
model is consistent (see Proposition 4.5).
Example 4.7. Let Γ be the dimer model given in Figure 2.1. We consider zigzag paths on Γ
given in Figure 3.1. We first collect the deformation data (see Definition 4.1). Let us choose the
zigzag path z3, and denote it by z. We see that ℓ(z) = 6, v := [z] = (−1,−1), and
∣∣ZI
v(Γ)
∣∣ = 1.
Since
∣∣ZI
v(Γ)
∣∣ = 1, we can take only r = 1, in which case h = ℓ(z)/2− r = 2. Thus, we consider
the deformation weight p = h = 2. The zig-deformation νzigp (Γ, z) of Γ at z with p = 2 is shown
in Figure 4.4.
(zig-1)
– (zig-3) (join)
Figure 4.4. The zig-deformation νzigp (Γ, z) of Γ at z.
We give the zag-deformation νzagp (Γ, z) of Γ at z with p = 2 as in Figure 4.5.
(zag-1)
– (zag-3) (join)
Figure 4.5. The zag-deformation νzagp (Γ, z) of Γ at z.
Example 4.8. We remark that even if Γ is an isoradial dimer model, the deformed ones are
not necessarily isoradial. For example, the leftmost dimer model Γ in Figure 4.6 is isoradial.
We choose a type I zigzag path z whose slope is v = (−1, 0), in which case
∣∣ZI
v(Γ)
∣∣ = 2 and
ℓ(z) = 4. We fix r = 1, and hence h = ℓ(z)/2− r = 1. Applying the zig-deformation at z with
p = 1 to Γ, we have the rightmost one in Figure 4.6. One can check that the deformed dimer
model is consistent but not isoradial.
z (zig-1)
– (zig-3) (join)
Figure 4.6. An example of the deformed dimer model that is not isoradial.
Example 4.9. We here give an example showing that the zig-deformation and zag-deformation
are not mutually inverse operations on the level of dimer models in general.
Deformations of Dimer Models 19
We consider the consistent dimer model Γ that takes the form as in the top-left of Figure 4.7,
and the type I zigzag path z on Γ. We see that [z] = (−1, 0) and ℓ(z) = 4. Let r = 1, and hence
h = ℓ(z)/2− r = 1. Then, the zig-deformation νzigp (Γ, z) of Γ at z with the deformation weight
p = h = 1 is the top-right of Figure 4.7. Through this process, the new zigzag path on νzigp (Γ, z)
whose slope is (1, 0) has been created. If we apply the zag-deformation with the deformation
weight q = 1 to this new zigzag path, then we recover the original dimer model Γ as we saw
in Proposition 4.6. On the other hand, there is another type I zigzag path on νzigp (Γ, z) whose
slope is (1, 0), which is denoted by z′. We see that ℓ(z′) = 4, and consider r = 1, and hence
h = ℓ(z′)/2− r = 1. Applying the zag deformation at z′ with the deformation weight q = h = 1
to νzigp (Γ, z), we have the dimer model νzagq
(
νzigp (Γ, z), z′
)
shown in the bottom-right of Figure 4.7,
which is not isomorphic to Γ. Nevertheless, Γ and νzagq
(
νzigp (Γ, z), z′
)
are “mutation-equivalent”
(see, e.g., [27, Section 5.11]) and hence their PM polygons are the same, see Appendix A for
more details of the mutation.
Γ :
: νzagq
(
νzigp (Γ, z), z′
)
z νzigp (−, z)
z′
(zag-1)
– (zag-3)
(join)
Figure 4.7. An example showing that the zig-deformation and zag-deformation are not mutually inverse
operations.
5 Combinatorial mutations of perfect matching polygons
In this section, we discuss a relationship between deformations of consistent dimer models and
combinatorial mutations of polygons. We first define combinatorial mutations for lattice poly-
topes of any dimension, and then we mainly discuss the case of polygons.
5.1 Preliminaries on combinatorial mutations of polytopes
Following [2], we introduce combinatorial mutations of polytopes. Let N ∼= Zd be a lattice of
rank d, and P ⊂ NR := N ⊗Z R be a convex lattice polytope. We assume that P contains the
origin 0. We denote by V(P ) the set of vertices of P . We say that two polytopes P,Q ⊂ NR are
isomorphic if they are transformed into each other by GL(d,Z)-transformations, in which case
we denote P ∼= Q.
We first prepare some notions used in the definition of combinatorial mutations of polytopes.
20 A. Higashitani and Y. Nakajima
Definition 5.1 (mutation data). Let P be a convex lattice polytope as above. In order to define
a combinatorial mutation of P , we fix the data consisting of a vector w and the associated integers
hmax(P,w), hmin(P,w) and width(P,w) defined as follows.
Let w ∈ M := HomZ(N,Z) ∼= Zd be a primitive lattice vector. The element w ∈ M determines
the linear map ⟨w,−⟩ : NR → R, where ⟨−,−⟩ is the natural inner product. We set
hmax(P,w) := max{⟨w, u⟩ |u ∈ P} and hmin(P,w) := min{⟨w, u⟩ |u ∈ P},
which are integers since P is a lattice polytope. When the situation is clear, we simply denote
these by hmax and hmin, respectively. We define the width of P with respect to w as
width(P,w) := hmax(P,w)− hmin(P,w).
We note that if the origin 0 is contained in the strict interior P ◦ of P , then hmin < 0 and
hmax > 0, in which case we have width(P,w) ≥ 2. We say that a lattice point u ∈ N (resp.
a subset F ∈ NR) is at height m with respect to w if ⟨w, u⟩ = m (resp. ⟨w, u⟩ = m for any
u ∈ F ).
For each height h ∈ Z, we let
wh(P ) := conv{u ∈ P ∩N | ⟨w, u⟩ = h},
which is the (possibly empty) convex hull of all lattice points in P at height h. By definition,
whmin
(P ) and whmax(P ) are faces of P . Using these notions, a combinatorial mutation of P is
defined as follows.
Definition 5.2. Let the notation be the same as above. We assume that there exists a lattice
polytope F ⊂ NR such that ⟨w, u⟩ = 0 for any u ∈ F , and for each negative height hmin ≤ h < 0
there exists a possibly empty lattice polytope Gh ⊂ NR satisfying
{u ∈ V(P ) | ⟨w, u⟩ = h} ⊆ Gh + (−h)F ⊆ wh(P ), (5.1)
where + means the Minkowski sum, and we especially define Q + ∅ = ∅ for any polytope Q.
We call F a factor of P with respect to w. Then, we define the combinatorial mutation of P
given by the vector w, factor F and polytopes {Gh} as
mutw(P, F ) := conv
( −1⋃
h=hmin
Gh ∪
hmax⋃
h=0
(wh(P ) + hF )
)
.
We note that the combinatorial mutation is independent of the choice of the polytopes {Gh}
(see [2, Proposition 1]). Also, a translation of the factor F does not affect the combinatorial
mutation; that is, for any u ∈ N with ⟨w, u⟩ = 0 we have mutw(P, F ) ∼= mutw(P, u+ F ); see [2]
for more details.
Remark 5.3 (the combinatorial mutation for the case of d = 2). When d = 2, we choose
an edge E of a lattice polygon P , and take w ∈ M ∼= Z2 as a primitive inner normal vector
for E. By the choice of w, we see that whmin
(P ) = E and whmax(P ) is either a vertex or an edge
of P . Then, we take a primitive lattice element uE ∈ N satisfying ⟨w, uE⟩ = 0, and define the
line segment F := conv{0, uE}, which is parallel to E at height 0 and has unit lattice length.
Since uE is uniquely determined up to sign, so is F . In this case, P admits a combinatorial
mutation with respect to w (equivalently, there exit polytopes {Gh} satisfying (5.1)) if and only
if |E ∩N | − 1 ≥ −hmin, see [22, Lemma 1]. We note that the combinatorial mutation does not
depend on the choice of uE (and hence F ). That is, mutw(P, F ) ∼= mutw(P,−F ), which means
they are GL(2,Z)-equivalent.
Deformations of Dimer Models 21
Example 5.4. We consider the polygon P given in the left of Figure 5.1 below, which coincides
with the PM polygon of the dimer model given in Figure 2.1 (see also Figure 2.5). Here, the
double circle stands for the origin 0. We consider the edge E whose primitive inner normal vector
is w = (1, 1), in which case hmin = −1 and hmax = 2. We take uE = (1,−1) ∈ N , which satisfies
⟨w, uE⟩ = 0, and consider the line segment F = conv{0, uE}. The combinatorial mutation
mutw(P, F ) of the polygon P is shown in the upper part of Figure 5.1. On the other hand, if we
consider the line segment −F = conv{0,−uE}, the combinatorial mutation mutw(P,−F ) is as
shown in the lower part of Figure 5.1.
mutw(−, F )
mutw(−,−F )
Figure 5.1. Examples of the combinatorial mutations of P .
Now, we collect fundamental properties of this combinatorial mutation.
Proposition 5.5 (see [2, Lemma 2 and Proposition 2]). Let the notation be the same as above.
(1) If Q := mutw(P, F ), then we have P = mut(−w)(Q,F ).
(2) P is a Fano polytope if and only if mutw(P, F ) is a Fano polytope.
Here, we recall that a convex lattice polytope P ⊂ NR with dimP = d is called Fano if the
origin is contained in the strict interior of P , and the vertices V(P ) of P are primitive lattice
points of N .
Then, we consider the combinatorial mutation of a lattice polytope P in terms of the polar
dual P ∗ of P in M . To do this, we must first discuss the polar dual P ∗ for a polyhedron P . We
consider the family of polyhedra (not necessarily convex polytopes) which are of the following
form:
Pd :=
{ ⋂
v∈S
Hv,≥−kv ∩
⋂
v′∈T
Hv′,≥0 ⊂ NR |S, T ⊂ M, |S|, |T | < ∞, kv ∈ Z>0
}
,
where Hv,≥k = {u ∈ NR | ⟨v, u⟩ ≥ k} for v ∈ M and k ∈ R. We note that any lattice polytope
containing the origin of NR belongs to Pd but the ones not containing the origin do not belong
to Pd since one of the supporting hyperplanes of such a polytope is of the form Hv,≥k for some
v ∈ M and some positive integer k. Similarly, we define Qd by swapping the roles of NR and MR.
For a given P ∈ Pd, we consider the polar dual P ∗ ⊂ MR of P defined as
P ∗ := {v ∈ MR | ⟨v, u⟩ ≥ −1 for all u ∈ P} ⊂ MR.
Then, we have the following statements.
22 A. Higashitani and Y. Nakajima
Proposition 5.6. Let the notation be the same as above. Then, for any P ∈ Pd we have
(i) P ∗ ∈ Qd,
(ii) (P ∗)∗ = P .
Proof. (i) Let P ∈ Pd. Since P is a polyhedron, there exist a polytope Q and a polyhedral
cone C such that P = Q+C, where + denotes the Minkowski sum (see [30, Corollary 7.1b]). Let
Q = conv
({
1
k1
u1, . . . ,
1
kp
up
})
and let C = cone({u′1, . . . , u′q}). Note that we can choose ui, u
′
j
from N and ki ∈ Z>0 because of the form of P . In what follows, we will show that
P ∗ =
p⋂
i=1
Hui,≥−ki ∩
q⋂
j=1
Hu′
j ,≥0.
First, we take v ∈ P ∗. Then, we have ⟨v, u⟩ ≥ −1 for any u ∈ P . Since 1
ki
ui ∈ Q + 0 ⊂ P ,
where 0 ∈ C denotes the origin, we see that ⟨v, 1
ki
ui⟩ ≥ −1, i.e., ⟨v, ui⟩ ≥ −ki for each i. If
there is j with ⟨v, u′j⟩ < 0, then ⟨v, u′+ ru′j⟩ < −1 for some u′ ∈ Q and some sufficiently large r.
Moreover, we have u′ + ru′j ∈ Q+ C = P . This contradicts v ∈ P ∗; thus ⟨v, u′j⟩ ≥ 0 for each j.
Therefore, v ∈
⋂p
i=1Hui,≥−ki ∩
⋂q
j=1Hu′
j ,≥0.
On the other hand, we take v ∈
⋂p
i=1Hui,≥−ki ∩
⋂q
j=1Hu′
j ,≥0. For any u ∈ P , as mentioned
above, there exist u′ ∈ Q and u′′ ∈ C such that u = u′ + u′′. Let u′ =
∑p
i=1
ri
ki
ui, where ri ≥ 0
with
∑p
i=1 ri = 1, and let u′′ =
∑q
j=1 sju
′
j , where sj ≥ 0. By using these expressions together
with the inequalities ⟨v, ui⟩ ≥ −ki for each i and ⟨v, u′j⟩ ≥ 0 for each j, we see that
⟨v, u⟩ = ⟨v, u′⟩+ ⟨v, u′′⟩ =
p∑
i=1
ri
ki
⟨v, ui⟩+
q∑
j=1
sj⟨v, u′j⟩ ≥ −
p∑
i=1
ri = −1,
and thus we have v ∈ P ∗.
(ii) For any u ∈ P , we have ⟨v, u⟩ ≥ −1 for any v ∈ P ∗, which means that P ⊂ (P ∗)∗. For
the other inclusion, we take u ∈ NR \ P . Let P =
⋂
v∈S Hv,≥−kv ∩
⋂
v′∈T Hv′,≥0. Then either
⟨v, u⟩ < −kv for some v ∈ S or ⟨v′, u⟩ < 0 for some v′ ∈ T holds. In the former case, since
⟨v, u′⟩ ≥ −kv for any u′ ∈ P , we have 1
kv
v ∈ P ∗. This means that there is v′′ := 1
kv
v ∈ P ∗ such
that ⟨v′′, u⟩ < −1, and hence u ̸∈ (P ∗)∗. Similarly, in the latter case, since ⟨rv′, u′⟩ ≥ 0 ≥ −1
for any u′ ∈ P and r ≥ 0, we have rv′ ∈ P ∗. This implies that u ̸∈ (P ∗)∗ for sufficiently large r,
and hence u ̸∈ (P ∗)∗. Therefore, we obtain (P ∗)∗ ⊂ P , as required. ■
We next define a map
φw,F : MR → MR as φw,F (v) := v − vminw,
where vmin := min{⟨v, u⟩ |u ∈ F}. In particular, when d = 2 (see Remark 5.3), this map can be
described as
φw,F (v) =
{
v if ⟨v, uE⟩ ≥ 0,
v − ⟨v, uE⟩w if ⟨v, uE⟩ < 0,
(5.2)
with F = conv{0, uE}. The next proposition is crucial to prove our main result Theorem 8.3.
Proposition 5.7. For any P ∈ Pd, we have
φw,F (P
∗) = mutw(P, F )∗.
Deformations of Dimer Models 23
Proof. Although this equality essentially follows from [2, Proposition 4] and the discussions in
[2, p. 12], we give a precise proof for completeness.
Let φ = φw,F and Q = mutw(P, F ). To show φ(P ∗) ⊂ Q∗, we take v ∈ P ∗ arbitrarily and
consider φ(v) = v − vminw ∈ φ(P ∗). We will show that ⟨v − vminw, u⟩ ≥ −1 for any u ∈ Q. It
suffices to show this for each vertex u ∈ V(Q).
� Let u ∈ V(Q) with ⟨w, u⟩ ≥ 0. Then we can write u = uP +⟨w, uP ⟩uF for some uP ∈ V(P )
and uF ∈ V(F ). In particular, we have
⟨v − vminw, u⟩ = ⟨v, uP ⟩+ ⟨w, uP ⟩(⟨v, uF ⟩ − vmin) ≥ ⟨v, uP ⟩ ≥ −1.
� Let u ∈ V(Q) with ⟨w, u⟩ < 0. For any uF ∈ V(F ), we have u − ⟨w, u⟩uF ∈ P . Hence,
⟨v, u− ⟨w, u⟩uF ⟩ ≥ −1. In particular, ⟨v, u⟩ ≥ −1 + vmin⟨w, u⟩. Thus, we see that
⟨v − vminw, u⟩ = ⟨v, u⟩ − vmin⟨w, u⟩ ≥ −1.
To show Q∗ ⊂ φ(P ∗), we will show that for any v ∈ Q∗ there is v′ ∈ P ∗ such that v = φ(v′).
Let ∆F be the normal fan of F in MR and let σ ∈ ∆F be a maximal cone in ∆F . The
discussions in [2, p. 12] say that there exists Mσ ∈ GLd(Z) such that the map φ is equal to Mσ,
i.e., φ(v) = vMσ. Thus, we conclude that φ(v′) = v for v′ = vM−1
σ . ■
Example 5.8. Let P be the polygon used in Example 5.4. The polar dual P ∗ takes the form as
in the left of Figure 5.2 below. We take w = (1, 1), uE = (1,−1) and consider the line segment
F = conv{0, uE} just like Example 5.4. Then, applying the piecewise-linear map (5.2) with uE
(resp. −uE) to the polar dual P ∗, we have the new polygon shown in the upper (resp. lower)
part of Figure 5.2. In this figure, the red line imply the points v ∈ R2 satisfying ⟨v, uE⟩ = 0.
We see that the resulting polygons are the polar dual of mutw(P, F ) and mutw(P,−F ) given in
Example 5.4 as shown in Proposition 5.7.
φw,F
φw,−F
Figure 5.2. Examples of the image of P ∗ under the piecewise-linear map (5.2).
5.2 The perfect matching polygons of deformed dimer models
We here consider the PM polygons of deformed dimer models.
24 A. Higashitani and Y. Nakajima
Example 5.9. Let Γ be the consistent dimer model given in Figure 2.1, and let νzigp (Γ, z)
(resp. νzagp (Γ, z)) be the zig-deformed (resp. zag-deformed) dimer model at z as shown in Fig-
ure 4.4 (resp. Figure 4.5) in Example 4.7. Here, we note the changes of zigzag paths on Γ under
these deformations. First, we recall the zigzag paths on Γ given in Figure 3.1.
z1 z2 z = z3 z4
Then, we observe zigzag paths on νzigp (Γ, z) and νzagp (Γ, z), but it would be more convenient
to see the zigzag paths on the dimer model shown in the middle of Figures 4.4 and 4.5 for
observing the changes of zigzag paths as in Figures 5.3 and 5.4. In both cases, the zigzag path
z = z3 on Γ is removed and the new zigzag paths, which are colored by red, are created. The
slopes of these new zigzag paths are −[z]. Oh the other hand, some slopes of zigzag paths on Γ
are preserved even if we apply the deformations. For example, we have the zigzag path whose
slope is [z2] on νzigp (Γ, z) which is colored by blue in Figure 5.3. Also, we have the zigzag paths
whose slopes are [z1], [z4] on νzagp (Γ, z) which are respectively colored by green and orange in
Figure 5.4. We will see these phenomena for more general situations in Propositions 7.5 and 7.6.
Figure 5.3. Some zigzag paths on the dimer model shown in the middle of Figure 4.4 which can be
reduced to νzigp (Γ, z).
Figure 5.4. Some zigzag paths on the dimer model shown in the middle of Figure 4.5 which can be
reduced to νzagp (Γ, z).
Then, we respectively have the PM polygons as in Figure 5.5. We see that the shape of the
PM polygon ∆
νzigp (Γ,z)
(resp. ∆νzagp (Γ,z)) coincides with the combinatorial mutation mutw(P, F )
(resp. mutw(P,−F )) given in Example 5.4.
Example 5.9 indicates a close connection between the PM polygon of a deformed dimer model
and the combinatorial mutation of the PM polygon of the original dimer model. In Section 8,
we will show this phenomenon for a general situation by considering the extended deformations
introduced in Section 6. In particular, as a special case of Theorem 8.3, we have Theorem 5.10
Deformations of Dimer Models 25
[z]
[z2] νzigp −[z]
[z2]
[z]
[z1]
[z4]
νzagp
−[z]
[z1]
[z4]
Figure 5.5. The changes of the associated PM polygon via the zig-deformation (upper) and the zag-
deformation (lower).
below (see also Remark 6.4 and Proposition 7.12), and this explains the phenomenon observed
in Example 5.9.
Theorem 5.10. Let the notation be the same as in Definitions 4.1, 4.3 and 4.4. We assume
that either one of the following conditions is satisfied:
(i) r = 1 or
(ii) Γ is a hexagonal or rectangular dimer model (see Definition 2.1).
Then we have
mutw(∆Γ, F ) = ∆
νzigp (Γ,{z1,...,zr}),
mutw(∆Γ,−F ) = ∆νzagp (Γ,{z1,...,zr})
for certain choices of the deformation data and the mutation data (see Setting 8.1).
Combining this result with some properties of the combinatorial mutation given in Sec-
tion 5.1, we have the following corollary, which will be generalized in Section 8 (see Corollaries 8.4
and 8.7).
Corollary 5.11. Using the same setting as in Theorem 5.10, we see that
� ∆
νzigp (Γ,{z1,...,zr})
∼= ∆νzagp (Γ,{z1,...,zr})
� ∆Γ is Fano if and only if ∆
νzigp (Γ,{z1,...,zr}) (resp. ∆νzagp (Γ,{z1,...,zr})) is Fano.
6 Extended deformations of dimer models
In this section, we add some procedures to the deformations given in Definitions 4.3 and 4.4
to construct a consistent dimer model having the same properties as Theorem 5.10 in a more
general situation. The operations which will be introduced in this section might be complicated,
thus we refer the reader to Figures 7.4–7.9 in the next section and Appendix B for recognizing
their points.
26 A. Higashitani and Y. Nakajima
Setting 6.1. Let Γ be a reduced consistent dimer model. As in Definition 4.1, we choose
a type I zigzag path z, and we let 2n := ℓ(z) and v := [z]. We fix positive integers r, h such that
r ≤
∣∣ZI
v(Γ)
∣∣ and n = r + h, and take a subset {z1, . . . , zr} ⊂ ZI
v(Γ) of type I zigzag paths. We
recall that we have 2n = ℓ(z1) = · · · = ℓ(zr) by Lemma 3.13, and hence we write zi as
zi = zi[1]zi[2] · · · zi[2n− 1]zi[2n].
Then we prepare the additional data as follows.
(1) We consider all zigzag paths x1, . . . , xs (resp. y1, . . . , yt) intersecting with z at some zags
(resp. zigs) of z. In this case, each of x1, . . . , xs (resp. y1, . . . , yt) intersects with any zi
at some zags (resp. zigs) of zi for all i = 1, . . . , r by Lemma 3.16. We may assume that
z1, . . . , zr are ordered cyclically in the sense that if xj (resp. yk) intersects with zi, then it
also intersects with zi−1 (resp. zi+1).
(2) We recall that |xj ∩ zi| (resp. |yk ∩ zi|) is the same number for any i = 1, . . . , r by
Lemma 3.16. Thus, for j ∈ {1, . . . , s} let mj := |xj ∩ zi| be this constant, and for
k ∈ {1, . . . , t} let m′
k := |yk ∩ zi|. We note that n = m1 + · · ·+ms = m′
1 + · · ·+m′
t.
(3) Then, we divide each zigzag path xj into mj parts x
(1)
j , . . . , x
(mj)
j as follows. We first fix
one of the intersections of zr and xj as the starting edge of x
(1)
j , and tracing along xj we
will arrive at another intersection of zr and xj . We consider the edge of xj just before this
intersection as the ending edge of x
(1)
j , and consider the second intersection as the starting
edge of x
(2)
j . Repeating this procedure, we obtain x
(1)
j , . . . , x
(mj)
j and the set of sub-zigzag
paths:{
x
(1)
1 , . . . , x
(m1)
1 , x
(1)
2 , . . . , x
(m2)
2 , . . . , x(1)s , . . . , x(ms)
s
}
. (6.1)
Similarly, we also divide each zigzag path yk into m′
k parts y
(1)
k , . . . , y
(m′
k)
k by considering
one of the intersections of z1 and yk as the starting edge of y
(1)
k , and obtain the set of
sub-zigzag paths:{
y
(1)
1 , . . . , y
(m′
1)
1 , y
(1)
2 , . . . , y
(m′
2)
2 , . . . , y
(1)
t , . . . , y
(m′
t)
t
}
. (6.2)
(4) We then assign one of {z1, . . . , zr} to x
(aj)
j for j = 1, . . . , s and aj = 1, . . . ,mj . Then, we
define Xi as the set of edges consisting of the intersections between zi and the sub-zigzag
paths in (6.1) that are assigned with zi. We can make this assignment so that |Xi| ≥ 1,
and then set pi := |Xi| − 1 for i = 1, . . . , r. We call X := {X1, . . . , Xr} a zig-deformation
parameter with respect to z1, . . . , zr and call the tuple p = (p1, . . . , pr) ∈ Zr
≥0 the weight
of X . Similarly, we assign one of {z1, . . . , zr} to y
(bk)
k for k = 1, . . . , t and bk = 1, . . . ,m′
k.
Then, we define Yi as the set of edges consisting of the intersections between zi and the
sub-zigzag paths in (6.2) that are assigned with zi. We can make this assignment so that
|Yi| ≥ 1, and then set qi := |Yi| − 1 for i = 1, . . . , r. We call Y := {Y1, . . . , Yr} a zag-
deformation parameter with respect to z1, . . . , zr and call the tuple q = (q1, . . . , qr) ∈ Zr
≥0
the weight of Y. We remark that p1 + · · ·+ pr = m1 + · · ·+ms − r = n− r = h, and also
that q1 + · · ·+ qr = h.
Definition 6.2 (extended zig-deformation). Let the notation be the same as in Setting 6.1.
For a zig-deformation parameter X = {X1, . . . , Xr} of weight p = (p1, . . . , pr), we consider the
operations (zig-1), (zig-2), (zig-3) defined in Definition 4.3 and use the same notation. Then we
conduct the following procedures:
Deformations of Dimer Models 27
(zig-4) For m = 1, . . . , n and i = 1, . . . , r, if the zag zi[2m] of the original zigzag path zi on Γ
is not contained in Xi, then we add edges, which we call bypasses (since these edges
provide a new route that connect edges of the zigzag path xj contained in non-deformed
parts, see Figure 7.9), connecting the following pairs of black and white nodes:
(wi,1[2m− 1], bi[2m+ 1]), (wi,2[2m− 1], bi,1[2m+ 1]),
. . . , (wi,pi [2m− 1], bi,pi−1[2m+ 1]), (wi[2m− 1], bi,pi [2m+ 1]).
We denote the resulting dimer mode by νzigX (Γ, {z1, . . . , zr}).
We note that νzigX (Γ, {z1, . . . , zr}) is non-degenerate by Proposition 7.1.
(zig-5) Then, we make the dimer model νzigX (Γ, {z1, . . . , zr}) consistent using the method given
in the proof of [3, Theorem 1.1] (see Operation 6.6 and Proposition 6.7).
At the end, we perform the operation (join) if the resulting dimer model contains 2-valent
nodes. We denote the resulting dimer model by νzigX (Γ, {z1, . . . , zr}), and call it the extended
zig-deformation of Γ at {z1, . . . , zr} with respect to the zig-deformation parameter X . If the
situation is clear, we simply denote this by νzigX (Γ).
zi[2m+ 1]
zi[2m]
zi[2m− 1]
xj
(zig-1)
– (zig-3) (zig-4)
Figure 6.1. The operations (zig-1)–(zig-4) at zi for the case pi = |Xi| − 1 = 2 and zi[m] ̸∈ Xi.
Similarly, we can define the “zag version” of this extended deformation as follows.
Definition 6.3 (extended zag-deformation). Let the notation be the same as in Setting 6.1.
For a zag-deformation parameter Y = {Y1, . . . , Yr} of weight q = (q1, . . . , qr), we consider the
operations (zag-1), (zag-2), (zag-3) defined in Definition 4.4 and use the same notation. Then
we conduct the following procedures:
(zag-4) For m = 1, . . . , n and i = 1, . . . , r, if the zig zi[2m − 1] of the original zigzag path zi
on Γ is not contained in Yi, then we add edges, which we call bypasses, connecting the
following pairs of black and white nodes:
(bi,1[2m], wi[2m+ 2]), (bi,2[2m], wi,1[2m+ 2]),
. . . , (bi,qi [2m], wi,qi−1[2m+ 2]), (bi[2m], wi,qi [2m+ 2]).
We denote the resulting dimer mode by νzagY (Γ, {z1, . . . , zr}).
We note that νzagY (Γ, {z1, . . . , zr}) is non-degenerate by Proposition 7.1.
(zag-5) Then, we make the dimer model νzagY (Γ, {z1, . . . , zr}) consistent using the method given
in the proof of [3, Theorem 1.1] (see Operation 6.6 and Proposition 6.7).
At the end, we perform the operation (join) if the resulting dimer model contains 2-valent
nodes. We denote the resulting dimer model by νzagY (Γ, {z1, . . . , zr}), and call it the extended
zag-deformation of Γ at {z1, . . . , zr} with respect to the zag-deformation parameter Y. If the
situation is clear, we simply denote this by νzagY (Γ).
28 A. Higashitani and Y. Nakajima
zi[2m+ 2]
zi[2m+ 1]
zi[2m]
yk
(zag-1)
– (zag-3) (zag-4)
Figure 6.2. The operations (zag-1)–(zag-4) at zi for the case qi = |Yi| − 1 = 2 and zi[2m+ 1] ̸∈ Yi.
Remark 6.4. If r = 1, then we consider deformation parameters X = {X1} and Y = {Y1}
in Setting 6.1, where X1 (resp. Y1) is the set of intersections between a chosen type I zigzag
path z and x1, . . . , xs (resp. y1, . . . , yt). In particular, X1 (resp. Y1) coincides with the set of zags
(resp. zigs) of z, and hence they are determined uniquely. Thus, in this case we may skip the
operations (zig-4) (resp. (zag-4)), in which case we may also skip (zig-5) (resp. (zag-5)), since
there are no bypasses (see also Observations 7.3, 7.4 and Lemma 7.7). Thus, in this case, the
extended deformations coincide with the usual deformations:
νzigX (Γ, {z}) = νzigp (Γ, {z}) and νzagY (Γ, {z}) = νzagq (Γ, {z}),
where p := |X1| − 1 = ℓ(z)/2− 1 and q := |Y1| − 1 = ℓ(z)/2− 1.
Remark 6.5. The non-degenerate dimer model νzigX (Γ, {z1, . . . , zr}) is determined uniquely for
given deformation data, but in the operation (zig-5), the procedure for removing edges is not
unique. Therefore, the resulting consistent dimer model is not unique, whereas, since the set of
slopes of zigzag paths is the same for all possible consistent dimer models (see Proposition 6.7(2)),
the associated PM polygon is the same by Proposition 3.6. In addition, it is generally believed
that all consistent dimer models associated with the same lattice polygon are transformed into
each other by the mutations of dimer models (see Appendix A). Thus, we expect that the
extended deformation of a consistent dimer model is determined uniquely up to “mutation
equivalence”. (We encounter the same situation for the extended zag-deformation.)
We observe that under the operations (zig-1)–(zig-4) (resp. (zag-1)–(zag-4)), (self-)intersec-
tions of zigzag paths may appear in the universal cover. We follow the strategy [3] to deal with
this as we explain now.
Operation 6.6. We note the operation given in the proof of [3, Theorem 1.1], which we use in
(zig-5) and (zag-5).
(a) The dimer model νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})), which is obtained by
applying the operations (zig-1)–(zig-4) (resp. (zag-1)–(zag-4)) to the reduced consistent
dimer model Γ, sometimes contains zigzag paths having self-intersections in the universal
cover. In this case, we use the operation given in the proof of [3, Theorem 1.1]; that is, we
remove all edges at self-intersections (see Figure 6.3). We note that this operation does
not change the slope of the argued zigzag path.
After this process, there might be a connected component of the resulting bipartite graph
that is contained in a simply-connected domain in T. In that case, we remove such a con-
nected component. We note that this removal does not affect our purpose, because our
main concern is the PM polygon which is recovered from the slopes of zigzag paths, and
the slope of the zigzag path corresponding to the argued connected component is trivial.
(b) On the other hand, the dimer model νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})) might
have a pair of zigzag paths on the universal cover that intersect with each other in the
Deformations of Dimer Models 29
Figure 6.3. An example of removing a self-intersection of a zigzag path.
same direction more than once. In this case, we use another operation given in the proof
of [3, Theorem 1.1], that is, we choose any such pair of zigzag paths and remove pairs of
consecutive intersections of this pair of zigzag paths (see Figure 6.4). We note that this
operation does not change the slopes of zigzag paths and the resulting bipartite graph is
also a dimer model because νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})) satisfies the
strong marriage condition, as we will see in Proposition 7.1.
Figure 6.4. An example of removing a pair of consecutive intersections of zigzag paths.
Since the dimer model νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})) is non-degenerate by
Proposition 7.1 and since it does not contain a homologically trivial zigzag path (see the proofs
of Propositions 7.5, 7.6 and 7.9), we can produce a dimer model satisfying the conditions in
Definition 3.2 from νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})) by iterated application of
Operation 6.6. Thus, νzigX (Γ, {z1, . . . , zr}) and νzagY (Γ, {z1, . . . , zr}) are consistent dimer models,
but those are not necessarily isoradial even if Γ is isoradial (see Example 4.8). Furthermore,
since the operation (join) does not change the slopes of zigzag paths, this proves the following
proposition.
Proposition 6.7. Let the notation be the same as Definitions 6.2 and 6.3. Then, we have the
following.
(1) The dimer models νzigX (Γ, {z1, . . . , zr}) and νzagY (Γ, {z1, . . . , zr}) are consistent.
(2) The set of slopes of zigzag paths on νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})) is the
same as that of νzigX (Γ, {z1, . . . , zr}) (resp. νzagY (Γ, {z1, . . . , zr})).
Remark 6.8. We note several additional points concerning the definition of the extended de-
formations.
(1) The join move does not change the slopes of zigzag paths, and hence it does not affect the
associated PM polygon. Thus, when we are interested in only the PM polygon, we may
skip (join).
(2) Even if we choose different sets of intersections X ′
1, . . . , X
′
r (resp. Y ′
1 , . . . , Y
′
r ) in Defini-
tion 4.1(7), the PM polygon of the deformed dimer model is the same as that of νzigX (Γ)
(resp. νzagY (Γ)), as we will show in Proposition 7.10.
30 A. Higashitani and Y. Nakajima
(3) If a dimer model Γ is hexagonal or rectangular (see Definition 2.1), we may skip the
operations (zig-4), (zig-5), (zag-4) and (zag-5) when we define νzigX (Γ, {z1, . . . , zr}) and
νzagY (Γ, {z1, . . . , zr}), as we will show in Proposition 7.12.
7 Foundations of extended deformations of dimer models
In this section, we observe the fundamental properties of extended zig-deformations and zag-
deformations. In particular, we will pursue the change of zigzag paths under these deformations
in Sections 7.2 and 7.3. We refer the reader to Appendix B for understanding those observation
in a concrete example.
Throughout this section, we keep the notation of Sections 4 and 6 unless otherwise stated.
7.1 The proof of the non-degeneracy
In this subsection, we prove the non-degeneracy of the dimer models νzigX (Γ, {z1, . . . , zr}) and
νzagY (Γ, {z1, . . . , zr}).
Proposition 7.1. The dimer models νzigX (Γ, {z1, . . . , zr}), and νzagY (Γ, {z1, . . . , zr}) are non-
degenerate.
Proof. We prove this for νzigX (Γ) = νzigX (Γ, {z1, . . . , zr}), the other case is similar. Let Γ′ be the
dimer model obtained by applying the operations (zig-1)–(zig-3) to Γ.
The first step. We recall that the non-degeneracy condition is equivalent to the strong mar-
riage condition; that is, a dimer model has equal numbers of black and white nodes and every
proper subset S of the black nodes satisfies the condition that S is connected to at least |S|+1
white nodes.
Suppose that Γ′ is non-degenerate. Then Γ′ satisfies the strong marriage condition. By
applying the operation (zig-4), we obtain the dimer model νzigX (Γ, {z1, . . . , zr}). Since (zig-4) is
the operation that adds new edges, it also satisfies the strong marriage condition, and hence it
is non-degenerate. Therefore, it is enough to show that Γ′ is non-degenerate.
The second step. We next consider a sub-dimer motel Γ′′ of Γ satisfying the condition (∗)
below. Note that Γ′′ is said to be a sub-dimer model of Γ if the set of the nodes coincide and
the set of edges in Γ′′ is a subset of edges in Γ.
Condition (∗): For any given edge e in Γ′′, let z′ and z′′ be the two different zigzag paths
on Γ′′ containing e. Then either (∗1) or (∗2) holds:
(∗1) Either [z′] ∈ {[zi],−[zi]} or [z′′] ∈ {[zi],−[zi]} holds;
(∗2) For any i, if z′ intersects with zi in Zig(zi) (resp. Zag(zi)), then z′′ intersects with zi
in Zag(zi) (resp. Zig(zi)).
A sub-dimer model Γ′′ of Γ satisfying (∗) can be constructed using the algorithm developed
in [14, 20] for proving Theorem 2.6. We adapt it to our situation.
First, let us consider the original dimer model Γ and let E1, E2, . . . , Em be all edges of ∆Γ,
where the primitive outer normal vector for E1 is the slope [z1] = · · · = [zr]. We assume that
these edges are ordered cyclically in the anti-clockwise direction (see Figure 8.1 as a reference).
Also, let vj be the primitive outer normal vector corresponding to Ej , for 1 ≤ j ≤ m. Let a
be the index such that va = −v1 if there exists such an edge among E2, . . . , Em; that is, Ea is
parallel to E1. If there is no such edge, then let Ea = ∅ for simplicity of notation. We recall
that by Proposition 3.5, for each Ej ̸= ∅ there exist zigzag paths on Γ such that the associated
slopes coincide with vj , and the set of such zigzag paths is denoted by Zvj = Zvj (Γ). By
our assumption, each slope of the zigzag paths in Z1 := Zv2 ∪ · · · ∪ Zva−1 and v1 are linearly
Deformations of Dimer Models 31
independent. Thus, the zigzag paths in Z1 intersect with a type I zigzag path zi precisely once
in the universal cover (see Lemma 3.15). By definition of E2, . . . , Ea−1, such an intersection
is given from the right of zi to the left of zi, and hence zigzag paths in Z1 intersect with zi
in Zag(zi). Similarly, we find that the zigzag paths in Z2 := Zva+1 ∪ · · · ∪ Zvm intersect with zi
precisely once in the universal cover, and specifically, they intersect with zi in Zig(zi).
If there are at least two edges between E2 and Ea−1 (i.e., a ≥ 4), then we take two adjacent
edges, say, E2 and E3. Since v2 and v3 are linearly independent, the zigzag paths z′2 ∈ Zv2 and
z′3 ∈ Zv3 intersect at some edge of Γ. Clearly, such an intersection z′2 ∩ z′3 is neither any edge
constituting any zigzag path whose slope is [zi] nor −[zi]. Now, remove an edge in z′2 ∩ z′3. This
operation merges z′2 and z′3, in which case the resulting dimer model stays consistent and the
associated PM polygon changes into the polygon obtained by cutting the corner of the original
PM polygon consisting of edges whose outer normal vectors are [z′2] and [z′3] (see [14, Sections 5
and 6] for more details). Furthermore, since z′2 ∩ zi ⊂ Zag(zi) and z′3 ∩ zi ⊂ Zag(zi) for each i,
edges in z′2∩z′3 do not share a node with zi for any i. Thus, zi is still type I even if we apply this
operation and the merged zigzag path intersects with zi in Zag(zi) . We repeat this procedure
until there are no two edges between E2 and Ea−1.
Similarly, if there are at least two edges between Ea+1 and Em (i.e., m−a ≥ 2), then we do the
same procedures as above until there are no two edges between Ea+1 and Em. After removing
all suitable edges from Γ, we get a consistent dimer model, which is clearly a sub-dimer model
of Γ, and we denote this by Γsub. Since we do not remove edges contained in a zigzag path
whose slope is ±[zi] in the above arguments, the edges E1 and Ea (if this is not empty) of ∆Γ
are preserved on ∆Γsub
(and hence we will use the same notation). Also, the edges E2, . . . , Ea−1
(resp. Ea+1, . . . , Em) of ∆Γ are substituted by a single edge in ∆Γsub
. We denote such an edge
by E′
2 (resp. E′
m). We note that zigzag paths corresponding to E′
2 (resp. E′
m) are obtained by
merging the ones corresponding to E2, . . . , Ea−1 (resp. Ea+1, . . . , Em). In particular, the PM
polygon ∆Γsub
is formed by E1, E
′
2, Ea, E
′
m, in which case ∆Γsub
is a triangle or a trapezoid.
Then, it follows from the construction of Γsub that
– the zigzag paths z1, . . . , zr on Γ are preserved on Γsub and they are type I;
– the zigzag paths on Γ corresponding to Ea (if this is not empty) are preserved on Γsub;
– the zigzag paths corresponding to E′
2 (resp. E′
m) are intersected with zi in Zag(zi) (resp.
Zig(zi)) (see also the argument in the proof of Lemma 3.14).
From these facts, it is easy to verify that Γsub is a sub-dimer model Γ′′ of Γ satisfying the
condition (∗).
The third step. Now, we apply the operations (zig-1)–(zig-3) in Definition 4.3 to Γsub, which is
possible since z1, . . . , zr are preserved on Γsub. We denote the resulting dimer model by Γ′
sub. By
construction, Γ′ can be obtained by adding some edges to Γ′
sub. Thus, similar to the discussion
in the first step, it is enough to show that Γ′
sub is non-degenerate to prove the non-degeneracy
of Γ′.
Thus, we now show that Γ′
sub is non-degenerate. To do this, we prove the existence of a perfect
matching that contains a given edge e of Γ′
sub. We divide the set of edges into four cases (i)–(iv):
(i) e is of the form (bi,j−1[2m − 1], wi,j [2m − 1]) for some 1 ≤ i ≤ r, 1 ≤ j ≤ pi + 1 and
1 ≤ m ≤ n, where we let bi,0[2m− 1] = bi[2m− 1] and wi,pi+1[2m− 1] = wi[2m− 1];
(ii) e is of the form (wi,j [2m− 1], bi,j [2m− 1]) for some 1 ≤ i ≤ r, 1 ≤ j ≤ pi and 1 ≤ m ≤ n;
(iii) e is of the form (bi,j [2m− 1], wi,j [2m− 3]) for some 1 ≤ i ≤ r, 1 ≤ j ≤ pi and 1 ≤ m ≤ n;
(iv) e is of a form other than those in (i)–(iii).
Namely, (i) and (ii) are the edges emanating from the process (zig-1), (iii) is an edge added in
the process (zig-3), and (iv) is an edge that is invariant between Γ′
sub and Γsub.
32 A. Higashitani and Y. Nakajima
Since Γsub is consistent and contains the type I zigzag paths z1, . . . , zr, there exist corner
perfect matchings P and P′ on Γsub that are adjacent and satisfy P ∩ zi = Zig(zi) and P′ ∩ zi =
Zag(zi) for 1 ≤ i ≤ r (see Section 3.2). Similarly, let Q be a corner perfect matching on Γsub
with Q ∩ zi = ∅ for 1 ≤ i ≤ r. The existence of such Q is guaranteed by Lemma 3.12. We will
use these P, P′ and Q in order to find a suitable perfect matching on Γ′
sub containing a given
edge e. We divide our discussions into the following cases (i)–(iv) that correspond to the above
division of edges.
Case (i): Let
P′′ =
(
P \
r⋃
i=1
Zig(zi)
)
∪
( r⋃
i=1
pi+1⋃
j=1
n⋃
m=1
(bi,j−1[2m− 1], wi,j [2m− 1])
)
; (7.1)
see Figure 7.1. It is easy to see that P′′ is a perfect matching on Γ′
sub containing e in the case (i).
Case (ii): Let
P′′ = Q ∪
( r⋃
i=1
pi⋃
j=1
n⋃
m=1
(wi,j [2m− 1], bi,j [2m− 1])
)
;
see Figure 7.2. Then, we see that P′′ is a perfect matching on Γ′
sub containing e in the case (ii).
Case (iii): Let
P′′ = Q ∪
( r⋃
i=1
pi⋃
j=1
n⋃
m=1
(bi,j [2m− 1], wi,j [2m− 3])
)
;
see Figure 7.3. Then, we see that P′′ is a perfect matching on Γ′
sub containing e in the case (iii).
Case (iv): We note that an edge e in the case (iv) also appears in Γsub, since e is unchanged
even if we apply (zig-1)–(zig-3). Thus, we can regard e as an edge of Γsub. Since Γsub satisfies
the condition (∗), the zigzag paths z′, z′′ on Γsub that contain e satisfy either (∗1) or (∗2).
� We assume that z′ and z′′ satisfy (∗1).
– Let, say, [z′] = [zi]. Since zigzag paths having the same slopes are obtained as the
difference of adjacent corner perfect matchings, either P or P′ contains e. If e ∈ P,
then we let P′′ be as in (7.1). Then, P′′ is a perfect matching on Γ′
sub containing e.
Even if e ∈ P′, we have the same conclusion by letting
P′′ =
(
P′ \
r⋃
i=1
Zag(zi)
)
∪
( r⋃
i=1
pi+1⋃
j=1
n⋃
m=1
(bi,j−1[2m− 1], wi,j [2m− 1])
)
.
– Let, say, [z′] = −[zi], in which case Ea ̸= ∅ and z′ corresponds to Ea. Let Q
′ and Q′′
be the corner perfect matchings on Γsub whose difference forms z′. Let e ∈ Q′. Since
h(Q′,P0) lies on Ea, where P0 is the reference perfect matching, we have Q′ ∩ zi = ∅
for any i by Lemma 3.9. Thus, we let
P′′ = Q′ ∪
( r⋃
i=1
pi⋃
j=1
n⋃
m=1
(wi,j [2m− 1], bi,j [2m− 1])
)
,
and see that P′′ is a perfect matching on Γ′
sub containing e.
Deformations of Dimer Models 33
� We assume that z′ and z′′ satisfy (∗2).
Let, say, z′ intersect with each zi in Zig(zi). Let Q
′ and Q′′ be the corner perfect matchings
on Γsub whose difference forms z′. Let e ∈ Q′. As noted above, we see that all zigzag paths
in Γsub intersecting with zi at some zig of zi have the same slopes. This implies that Q′
contains all zigs of zi, i.e., Q
′ ∩ zi = Zig(zi). This also means that Q′ = P. Hence, we
let P′′ be the same as (7.1) and see that P′′ is a perfect matching containing e. ■
(zig-1)–(zig-3)
Figure 7.1. The perfect matchings P on Γsub (left) and P′′ on Γ′
sub (right) for the case (i).
(zig-1)–(zig-3)
Figure 7.2. The perfect matchings Q on Γsub (left) and P′′ on Γ′
sub (right) for the case (ii).
(zig-1)–(zig-3)
Figure 7.3. The perfect matchings Q on Γsub (left) and P′′ on Γ′
sub (right) for the case (iii).
7.2 Behaviors of zigzag paths after extended deformations
In this subsection, we study zigzag paths of the deformed dimer models and their slopes. We
mainly discuss the extended zig-deformation, but the same assertions hold for the extended
zag-deformation by a similar argument. We will work with the notation in Definition 4.1 and
Setting 6.1. We consider the extended deformation νzigX (Γ, {z1, . . . , zr}) of Γ (see Definitions 4.3
and 6.2).
First, we observe zigzag paths of Γ and fix the notation which we will use throughout this
section.
Observation 7.2. Let z1, . . . , zr be type I zigzag paths of Γ with [z1] = · · · = [zr]. These zigzag
paths are ordered cyclically along the subscript i = 1, . . . , r. For any α ∈ Z and i = 1, . . . , r,
34 A. Higashitani and Y. Nakajima
let z̃i(α) be a zigzag path on the universal cover Γ̃ whose projection on Γ is zi. Each z̃i(α)
divides R2 into two parts, and thus it makes sense to consider the left of z̃i(α) and the right
of z̃i(α). Then, we can write a straight line ℓLi,α (resp. ℓRi,α) on the left (resp. right) of z̃i(α) such
that the gradient of ℓLi,α (resp. ℓRi,α) is v = [zi] and the nodes contained in the region obtained as
the intersection of the right of ℓLi,α and the left of ℓRi,α are precisely the nodes located on z̃i(α).
We will call such a region the (i, α)-th deformed part (see Figure 7.4).
z̃i(α)
ℓLi,α
ℓRi,α
the left of ℓLi,α
the right of ℓRi,α
the (i, α)-th
deformed part
Figure 7.4.
Also, we call the region obtained as the intersection of the right of ℓRi−1,α and the left of ℓLi,α
the (i, α)-th irrelevant part. Here, we say that the intersection of the right of ℓRr,α and the left
of ℓL1,α+1 is the (1, α + 1)-th irrelevant part. We remark that sometimes there are no nodes in
an irrelevant part. We sometimes omit α ∈ Z from the notation unless it causes confusion. We
also use these terminologies for Γ. That is, a part of Γ obtained by projecting a deformed (resp.
irrelevant) part of Γ̃ onto Γ is said to be a deformed (resp. irrelevant) part of Γ.
the (i, α)-th
irrelevant part
the (i, α)-th
deformed part
the (i+ 1, α)-th
irrelevant part
the (i+ 1, α)-th
deformed part
the (i+ 2, α)-th
irrelevant part
the (i+ 2, α)-th
deformed part
the (i+ 3, α)-th
irrelevant part
z̃i(α) z̃i+1(α) z̃i+2(α)
Figure 7.5.
By the condition of Definition 3.3(3), z1, . . . , zr do not have a common node, and therefore
the irrelevant parts do not overlap each other. Since the operations (zig-1)–(zig-4) (or (zag-1)–
(zag-4)) are local operations on each deformed part, any irrelevant part will be unchanged even
if we apply these operations. Thus, we continue to use these terminologies “deformed parts”
and “irrelevant parts”. We then consider a zigzag path w satisfying the following properties:
Deformations of Dimer Models 35
(a) If [zi] and [w] are linearly independent, then by Lemma 3.15, w̃ intersects with z̃i(α)
precisely once, and so does any z̃i(α) with i = 1, . . . , r and α ∈ Z. In particular, all
intersections are zigs of w or zags of w by Lemma 3.16. If w̃ intersects with z̃i(α) at
a zig (resp. zag) of z̃i(α), then we easily see that w̃ crosses the (i, α)-th deformed part
in the direction from the (i, α)-th (resp. (i + 1, α)-th) irrelevant part to the (i + 1, α)-th
(resp. (i, α)-th) irrelevant part.
(b) If [zi] and [w] are linearly dependent, then by Lemma 3.15, w̃ and z̃i(α) do not intersect
for any i = 1, . . . , r and α ∈ Z. This is equivalent to the condition that w̃ is contained
in some irrelevant part. In this case, w is unchanged even if we apply the extended
deformations because (zig-1)–(zig-4) (or (zag-1)–(zag-4)) are operations on the deformed
parts, and (zig-5) (or (zag-5)) does not affect w by Lemma 7.7 below.
In the remainder of this subsection, we discuss the behavior of zigzag paths after applying the
operations (zig-1)–(zig-4). (We can apply the same arguments for the case of (zag-1)–(zag-4).)
Observation 7.3. We consider a zigzag path yk on a consistent dimer model Γ intersecting
with zi at a zig of zi. Let z̃i and ỹk be zigzag paths on Γ̃ projecting onto zi and yk, respectively.
By Observation 7.2(a), ỹk crosses the i-th deformed part in the direction from the i-th irrelevant
part to the (i+ 1)-th irrelevant part, and ỹk intersects with z̃i precisely once. We suppose that
the zig z̃i[2m− 1] of z̃i is such an intersection. In this case, z̃i[2m− 1] is also a zag of ỹk; thus
we may write it as ỹk[2m].
Now, we apply the operations (zig-1)–(zig-3) to Γ, and we denote the resulting dimer model
by Γ′ and its universal cover by Γ̃′. Then, some new nodes are inserted in z̃i[2m−1] = ỹk[2m], and
zigzag paths z̃i,1, . . . , z̃i,pi on Γ̃′, which project onto zigzag paths zi,1, . . . , zi,pi on Γ′, respectively,
appear in the i-th deformed part.
ỹkz̃i
(zig-1)–(zig-3)
ỹ′k
z̃i,1 z̃i,pi
Figure 7.6.
We consider the zigzag path ỹ′k on Γ̃′ passing through ỹk[2m− 1] as a zig. That is, ỹ′k starts
from ỹk[2m−1], crosses through zigzag paths z̃i,1, . . . , z̃i,pi in the i-th deformed part, and arrives
at ỹk[2m + 1] (see the right of Figure 7.6). In particular, it crosses the i-th deformed part in
the direction from the i-th irrelevant part to the (i + 1)-th irrelevant part. Although ỹ′k looks
different from ỹk in the deformed parts, it connects the zigs ỹk[2m−1] and ỹk[2m+1] of ỹk in the
i-th deformed part for all i, and thus ỹ′k shares the same nodes and edges as ỹk in any irrelevant
part. We conclude that ỹ′k coincides with ỹk in all irrelevant parts, and behaves as depicted on
the right-hand side of Figure 7.6 in each deformed part. Also, we see that bypasses inserted in
the operation (zig-4) do not affect the behavior of ỹ′k, because ỹ
′
k never passes through bypasses.
Observation 7.4. We consider a zigzag path xj on Γ intersecting with zi at a zag of zi. Let x̃j be
a zigzag path on Γ̃ projecting onto xj . By Observation 7.2(a), x̃j crosses the i-th deformed part
in the direction from the (i+ 1)-th irrelevant part to the i-th irrelevant part, and x̃j intersects
with z̃i precisely once. We suppose that the zag z̃i[2m] of z̃i is such an intersection. In this case,
z̃i[2m] is also a zig of x̃j , and thus we may write it as x̃j [2m+1]. We then apply the operations
(zig-1)–(zig-4) to Γ, and we have the dimer model νzigX (Γ) = νzigX (Γ, {z1, . . . , zr}).
36 A. Higashitani and Y. Nakajima
If zi[2m], which is the projection of z̃i[2m] = x̃j [2m + 1] on Γ, is not contained in Xi, then
bypasses are inserted. We consider the zigzag path x̃′j on the universal cover of νzigX (Γ) passing
through x̃j [2m] as a zag of x̃′j . That is, x̃
′
j starts from x̃j [2m], behaves as depicted on the right-
hand side of Figure 7.7, and arrives at x̃j [2m + 2]. In particular, x̃′j crosses the i-th deformed
part in the direction from the (i+ 1)-th irrelevant part to the i-th irrelevant part, and behaves
in the same manner as x̃j in each irrelevant part.
z̃i
x̃j
(zig-1)–(zig-4)
x̃′j
z̃i,1 z̃i,pi
Figure 7.7. The case where the projection of z̃i[2m] = x̃j [2m+ 1] on Γ is not contained in Xi.
z̃i
x̃j
(zig-1)–(zig-4)
x̃′j
z̃i,1 z̃i,pi
Figure 7.8. The case where the projection of z̃i[2m] = x̃j [2m+ 1] on Γ is contained in Xi.
If zi[2m] is contained in Xi, then no bypasses are inserted. We again consider the zigzag
path x̃′j on the universal cover of νzigX (Γ) passing through x̃j [2m] as a zag of x̃′j (e.g., see
Figure 7.8). Unlike in the previous case, after passing through x̃j [2m], x̃′j goes to the edge{
b̃i[2m + 1], w̃i,1[2m + 1]
}
. (Here, we denote the edge whose endpoints are a black node b and
a white node w by {b, w}.)
If zi[2m+2] ∈ Xi, in which case bypasses are not inserted, then x̃′j goes to the edge
{
w̃i,1[2m+
1], b̃i,1[2m+ 3]
}
.
On the other hand, if zi[2m+ 2] ̸∈ Xi, in which case we insert bypasses, then x̃′j goes to the
edge
{
w̃i,1[2m+1], b̃i[2m+3]
}
. In such a way, x̃′j crosses the i-th deformed part in the direction
from the (i+1)-th irrelevant part to the i-th irrelevant part. More precisely, if we assume that x̃′j
goes through the edge {b̃i,s[2m− 1], w̃i,s+1[2m− 1]} in the i-th deformed part, then x̃′j behaves
as follows:
(1) if zi[2m] ∈ Xi, in which case bypasses are not inserted, then x̃′j goes through the edge{
w̃i,s+1[2m− 1], b̃i,s+1[2m+ 1]
}
and then
{
b̃i,s+1[2m+ 1], w̃i,s+2[2m+ 1]
}
,
(2) if zi[2m] ̸∈ Xi, in which case we insert bypasses, then x̃′j goes through the edge
{
w̃i,s+1[2m−
1], b̃i,s[2m+ 1]
}
and then
{
b̃i,s[2m+ 1], w̃i,s+1[2m+ 1]
}
,
where m = 1, . . . , n and s = 0, . . . , pi − 1 with b̃i,0[−] = b̃i[−] and w̃i,pi+1[−] = w̃i[−]. When we
consider x̃′j in the i-th deformed part, we encounter case (1) |Xi| times and case (2) ℓ(zi)/2−|Xi|
Deformations of Dimer Models 37
times. Since pi = |Xi|−1, we see that x̃′j goes out the i-th deformed part from w̃i[2m−1+2n] =
w̃i[2m− 1 + ℓ(zi)], and then it goes into the i-th irrelevant part. Thus, x̃′j behaves in the same
manner as the shift of x̃j in the i-th irrelevant part. For example, if we consider the type I zigzag
path zi with ℓ(zi) = 8, and the zig-deformation parameter Xi with |Xi| = 3 and zi[2m+4] ̸∈ Xi,
then the i-th deformed part will change as shown in Figure 7.9.
x̃j
x̃j(1)
z̃i[2m− 1]
z̃i[2m]
z̃i[2m+ 1]
z̃i[2m+ 7]
w̃i[2m− 1]
w̃i[2m+ 7]
b̃i[2m− 1]
b̃i[2m+ 7]
(zig-1)–(zig-4)
x̃′
j
w̃i[2m− 1]
w̃i[2m+ 7]
b̃i[2m− 1]
b̃i[2m+ 7]
Figure 7.9. An example of the behavior of x̃′
j in the i-th deformed part.
7.3 Properties of zigzag paths on deformed dimer models
In this subsection, we study the slopes of zigzag paths on deformed dimer models. In particular,
we can describe them in terms of zigzag paths of the original dimer model, and this description
plays a crucial role when discussing the relationship with the combinatorial mutations of the
associated polygons.
Proposition 7.5. The path zi,j given in (zig-3) of Definition 4.3 (resp. (zag-3) of Definition 4.4)
is a type I zigzag path zi,j of νzigX (Γ) (resp. νzagY (Γ)) with ℓ(zi,j) = ℓ(zi). Moreover, zi,j does not
have any self-intersections on the universal cover, and satisfies [zi,j ] = −[zi] = −v, and hence it
is not homologically trivial.
Proof. We consider the case of νzigX (Γ). The case of νzagY (Γ) is similar.
First, zi,j is a zigzag path of νzigX (Γ) by definition. We see that zigzag paths of νzigX (Γ) inter-
secting with z̃i,j take either the form of ỹ′k given in Observation 7.3 or x̃′j given in Observation 7.4.
In particular, these intersect with z̃i,j precisely once in each deformed part, and ỹ′k crosses the
i-th deformed part in the direction from the i-th irrelevant part to the (i+1)-th irrelevant part,
and x̃′j crosses the i-th deformed part in the direction from the (i+ 1)-th irrelevant part to the
i-th irrelevant part for all i. Since other zigzag paths do not intersect with z̃i,j , we see that zi,j
is a type I zigzag path of νzigX (Γ). Also, z̃i,j does not have any self-intersections by definition.
By these properties, the edges constituting zi,j are not removed by the operation (zig-5). In ad-
dition, since zi,j contains no 2-valent nodes, the operation (join) does not affect zi,j . Therefore,
we see that zi,j is a type I zigzag path of νzigX (Γ), and the remaining assertions follow from the
definition of zi,j . ■
Proposition 7.6. The paths y1, . . . , yt (resp. the paths x1, . . . , xs) are zigzag paths of Γ inter-
secting with a chosen type I zigzag path z at some zigs (resp. zags) of z. We have the following.
38 A. Higashitani and Y. Nakajima
(1) For any zigzag path yk of Γ, there exists a unique zigzag path y′k of νzigX (Γ) without self-
intersections on the universal cover and satisfying [yk] = [y′k] where k = 1, . . . , t.
(2) For any zigzag path xj of Γ, there exists a unique zigzag path x′j of νzagY (Γ) without self-
intersections on the universal cover and satisfying [xj ] = [x′j ] where j = 1, . . . , s.
Proof. We consider the case of νzigX (Γ). The case of νzagY (Γ) is similar.
We use the same notation used in Observation 7.3. In particular, we consider the zigzag
path ỹ′k of Γ̃′ which coincides with ỹk in all irrelevant parts, and behaves as depicted on the
right-hand side of Figure 7.6 in each deformed part, and therefore does not have any self-
intersections. Since bypasses inserted in the operation (zig-4) do not affect the behavior of ỹ′k,
we can extend ỹ′k to a zigzag path of the universal cover of νzigX (Γ). By projecting ỹ′k onto νzigX (Γ),
we have the zigzag path y′k of νzigX (Γ). By the construction given in Observation 7.3, we see that
[yk] = [y′k]. Since ỹ
′
k coincides with ỹk in all irrelevant parts and Γ is consistent, it does not behave
pathologically in irrelevant parts as it infringes on the consistency condition. Furthermore, ỹ′k
intersects with zigzag paths of the forms z̃i,j and x̃′j (see Observation 7.4) in some deformed parts,
but they do not intersect with each other in the same direction more than once. Therefore, the
edges constituting y′k are not removed by the operation (zig-5), and hence y′k is not changed
by (zig-5). In addition, the operation (join) does not change the slopes. Thus, we naturally
extend this zigzag path y′k as the path of νzigX (Γ), which is determined uniquely and satisfies
[yk] = [y′k] by construction. ■
By the proof of Propositions 7.5 and 7.6, the zigzag paths z̃i,j and ỹ′k do not have any self-
intersections, and do not intersect with other zigzag paths in the same direction more than once.
Furthermore, the intersections between x̃′j and z̃i,j or ỹ′k are not bypasses (see Observations 7.3
and 7.4). This proves the following lemma.
Lemma 7.7. We have the following.
(1) The edges removed by the operation (zig-5) are a part of the bypasses added in (zig-4) or
edges appearing in some irrelevant parts that are intersections between pairs of zigzag paths
x1, . . . , xs.
(2) The edges removed by the operation (zag-5) are a part of the bypasses added in (zag-4)
or edges appearing in some irrelevant parts that are intersections between pairs of zigzag
paths y1, . . . , yt.
Before showing the next proposition, we introduce some notation.
Setting 7.8. Let z be a zigzag path on a consistent dimer model Γ. We recall that corner perfect
matchings are ordered in the anti-clockwise direction along the vertices of ∆Γ (see Section 2.2).
Let P, P′ be adjacent corner perfect matchings on Γ such that the difference of P and P′ contains z
(see Proposition 3.5). We assume that P, P′ are ordered with this order, in which case P ∩
z = Zig(z) and P′ ∩ z = Zag(z). Then, we set Pz := P and P′
z := (P\Zig(z)) ∪ Zag(z). By
Proposition 3.6, Pz, P
′
z are boundary perfect matchings corresponding to certain lattice points
on the edge of ∆Γ whose outer normal vector is [z], and we see that the difference of Pz and P′
z,
namely Pz∪P′
z\Pz∩P′
z, forms z. Thus, by this construction, h(P′
z,Pz) ∈ Z2 is a primitive lattice
element with ⟨[z], h(P′
z,Pz)⟩ = 0.
Proposition 7.9. Let h(P′
z,Pz) be a primitive lattice element as above. We have the following.
(1) For any zigzag path xj of Γ, there exists a unique zigzag path x′j of νzigX (Γ) such that for
every j = 1, . . . , s
[x′j ] = [xj ] + ⟨[xj ], h(P′
z,Pz)⟩[z].
Deformations of Dimer Models 39
(2) For any zigzag path yk of Γ, there exists a unique zigzag path y′k of νzagY (Γ) such that for
every k = 1, . . . , t
[y′k] = [yk] + ⟨[yk], h(P′
z,Pz)⟩[z].
Proof. We consider the case of νzigX (Γ). The case of νzagY (Γ) is similar.
We recall that |xj ∩ z| = |xj ∩ zi| for all i = 1, . . . , r, and that this number is denoted by mj
(see Definition 4.1). We first show that
mj = ⟨[xj ], h(P′
z,Pz)⟩ (7.2)
for j = 1, . . . , s. Let pxj be the path of the quiver QΓ going along the left side of xj (see
Observation 3.7). In particular, considering pxj as an element in H1(T), we have [pxj ] = [xj ].
By our assumption, the intersections xj ∩z are contained in Zag(z) = P′∩z. Thus, pxj crosses z
at a zig of z. Since P ∩ z = Zig(z), every time pxj crosses z, the height function hP′,P increases
by 1. Since mj = |xj ∩ z|, we have the equation (7.2).
Then, we show that for each xj where j = 1, . . . , s, there exists a zigzag path x′j on νzigX (Γ) =
νzigX (Γ, {z1, . . . , zr}) such that
[x′j ] = [xj ] +mj [z]. (7.3)
We divide xj into sub-zigzag paths x
(1)
j , . . . , x
(mj)
j . By definition, x
(1)
j intersects with zr at a zag
of zr. We denote this zag by zr[2m] := x
(1)
j ∩zr, in which case the white (resp. black) node that is
the end point of zr[2m] is denoted by wr[2m−1] (resp. br[2m+1]). By considering the universal
cover Γ̃, we naturally define x̃j , x̃
(1)
j , z̃r, z̃r[2m], w̃r[2m − 1], b̃r[2m + 1], etc. Also, we assume
that z̃r is contained in the (r, 0)-th deformed part. Then, x̃j crosses the (r, 0)-th deformed part
in the direction from the (r + 1, 0)-th irrelevant part to the (r, 0)-th irrelevant part, in which
case the entrance of the (r, 0)-th deformed part is b̃r[2m+ 1] and the exit is w̃r[2m− 1].
In what follows, we use the same notation used in Observation 7.4. In particular, we pay
attention to the zigzag path x̃′j of the universal cover νzigX (Γ)∼ of νzigX (Γ), which behaves as
follows:
(Ar) If zr[2m] ̸∈ Xr, then x̃′j goes into the (r, 0)-th deformed part of νzigX (Γ)∼ from b̃r[2m + 1]
and goes out from w̃r[2m− 1] (see also Figure 7.7). After crossing the (r, 0)-th deformed
part, it goes into the (r, 0)-th irrelevant part, and it behaves in the same manner as x̃j in
that part.
(Br) If zr[2m] ∈ Xr, then x̃′j goes into the (r, 0)-th deformed part of νzigX (Γ)∼ from b̃r[2m + 1]
and goes out from w̃r[2m − 1 + 2n] = w̃r[2m − 1 + ℓ(z)] (see also Figures 7.8 and 7.9).
After crossing the (r, 0)-th deformed part, it goes into the (r, 0)-th irrelevant part, and it
behaves in the same manner as the shift of x̃j , which we denote as x̃j(1), in that part.
Then, x̃′j goes into the (r−1, 0)-th deformed part of νzigX (Γ)∼. We let z̃r−1[2m
′] := x̃j ∩ z̃r−1 and
z̃r−1[2m
′′] := x̃j(1)∩ z̃r−1. We note that on the dimer model Γ we have zr−1[2m
′] = zr−1[2m
′′] =
x
(1)
j ∩ zr−1 by definition.
� If zr[2m] ∈ Xr in the above argument, then zr−1[2m
′] ̸∈ Xr−1 by the definition of Xr
and Xr−1. (Furthermore, x
(1)
j ∩ zi ̸∈ Xi for any i ̸= r.) In this case, x̃′j crosses the
(r − 1, 0)-th deformed part of νzigX (Γ)∼ in the same manner as (Ar) above. Then, it
behaves in the same manner as x̃j(1) in the (r − 1, 0)-th irrelevant part.
� Let zr[2m] ̸∈ Xr in the above argument.
40 A. Higashitani and Y. Nakajima
– If zr−1[2m
′] ̸∈ Xr−1, then x̃′j crosses the (r− 1, 0)-th deformed part of νzigX (Γ)∼ in the
same way as (Ar). Then, it behaves in the same manner as x̃j in the (r − 1, 0)-th
irrelevant part.
– If zr−1[2m
′] ∈ Xr−1, then x̃′j crosses the (r− 1, 0)-th deformed part of νzigX (Γ)∼ in the
same way as (Br). Then, it behaves in the same manner as x̃j(1) in the (r − 1, 0)-th
irrelevant part.
Repeating these inductive arguments, we see that x̃′j crosses the (i, 0)-th deformed part of
νzigX (Γ)∼ for i = r, r − 1, . . . , 1 in this order, and goes into the (1, 0)-th irrelevant part. In any
case, x̃′j behaves in the same manner as x̃j(1) in this irrelevant part.
Then, x̃′j goes into the (r,−1)-th deformed part, in which case we consider the sub-zigzag
path x
(2)
j of Γ and the intersection between z̃r(−1) and x̃
(2)
j on Γ̃. By the same arguments as
above, we see that x̃′j crosses the (i,−1)-th deformed part of νzigX (Γ)∼ for i = r, r − 1, . . . , 1 in
this order. After that, it goes into (1,−1)-th irrelevant part and behaves in the same manner
as x̃j(2) in this irrelevant part.
Repeating these arguments, we finally see that x̃′j crosses the (i,−mj + 1)-th deformed part
of νzigX (Γ)∼ for i = r, r− 1, . . . , 1 in this order, and behaves in the same manner as x̃j(mj) in the
(1,−mj + 1)-th irrelevant part. Then, x̃′j goes into the (r,−mj)-th deformed part of νzigX (Γ)∼,
in which case we denote the black node that is the entrance of this deformed part by B. Since
mj = |xj ∩ zi|, the projection of x̃j ∩ z̃i(−mj) on Γ coincides with zi[2m], which is the starting
edge of our arguments. Thus, B coincides with b̃r[2m + 1] if they are projected onto νzigX (Γ),
which means we can follow all edges of the zigzag path x′j of νzigX (Γ). By these arguments, we
see that the slope of x̃′j changes [z] in each deformed part, and thus we have [x′j ] = [xj ] +mj [z].
Finally, we apply the operations (zig-5) and (join) to νzigX (Γ). Then, we obtain the deformed
dimer model νzigX (Γ) and the zigzag path on it having the same slope as x̃′j . This zigzag path
is determined uniquely by the construction, and we use the same notation for this zigzag path
by abuse of the notation. Since (zig-5) and (join) do not change the slopes of zigzag paths, we
have (7.3). ■
Since the slopes of zigzag paths νzigX (Γ) and νzagY (Γ) do not depend on the choice of X1, . . . , Xr
(resp. Y1, . . . , Yr) by Propositions 7.5, 7.6 and 7.9, we obtain the proposition below. However,
we remark that the deformed dimer model depends on the choice of zig-deformation parameters;
thus νzigX (Γ, {z1, . . . , zr}) ̸∼= νzigX ′ (Γ, {z1, . . . , zr}) in general (the zag version is similar).
Proposition 7.10. For zig-deformation parameter X ′ := {X ′
1, . . . , X
′
r} and zag-deformation
parameter Y ′ := {Y ′
1 , . . . , Y
′
r} different from X and Y, we have
∆
νzigX (Γ,{z1,...,zr}) = ∆
νzigX′ (Γ,{z1,...,zr})
and ∆νzagY (Γ,{z1,...,zr}) = ∆νzagY′ (Γ,{z1,...,zr}).
7.4 Remarks on the extended deformations of hexagonal
and rectangular dimer models
As we mentioned in Remark 6.8, we can skip the operations (zig-4) and (zig-5) (resp. (zag-4)
and (zag-5)) in the case of hexagonal and rectangular dimer models (see Definition 2.1), as we
will see below. We note that hexagonal and rectangular dimer models are isoradial. These
dimer models have been studied in several papers, with the following results being particularly
noteworthy.
Proposition 7.11 (e.g., [21, 28, 31]). Let Γ be a consistent dimer model. Then, we have the
following.
Deformations of Dimer Models 41
(1) Γ is a hexagonal dimer model if and only if the PM polygon ∆Γ is a triangle.
(2) If Γ is a rectangular dimer model, then the PM polygon ∆Γ is a parallelogram.
For these nice classes of dimer models, we may skip the operations (zig-4) and (zig-5) (or
(zag-4) and (zag-5)) when we apply the extended deformation.
Proposition 7.12. Let Γ be a hexagonal or rectangular dimer model. Then, the extended
deformation νzigX (Γ, {z1, . . . , zr}) is defined by the operations (zig-1)–(zig-3) and (join). Similarly,
the extended deformation νzagY (Γ, {z1, . . . , zr}) is defined by the operations (zag-1)–(zag-3) and
(join).
In particular, the extended deformations coincide with the usual deformations:
νzigX (Γ, {z1, . . . , zr}) = νzigp (Γ, {z1, . . . , zr}) and
νzagY (Γ, {z1, . . . , zr}) = νzagq (Γ, {z1, . . . , zr}).
Proof. We will prove the case of the extended zig-deformation. The case of the extended
zag-deformation is similar.
The zigzag paths z1, . . . , zr have the same slope, and this slope is the outer normal vector of
an edge of the PM polygon ∆Γ by Proposition 3.5.
(1) Let Γ be a hexagonal dimer model. Then, ∆Γ is a triangle by Proposition 7.11(1). Let e1,
e2, e3 be the edges of ∆Γ ordered cyclically in the anti-clockwise direction. We may assume
that the slopes of z1, . . . , zr are the outer normal vector of e1. We then consider the zigzag
paths x1, . . . , xs (resp. y1, . . . , yt) intersecting with zi at zags (resp. zigs) of zi. Since Γ is
isoradial, it is properly ordered. Thus, by Proposition 3.5 we see that [x1] = · · · = [xs]
(resp. [y1] = · · · = [yt]), and [xj ] (resp. [yk]) is the outer normal vector of e2 (resp. of e3).
(2) Let Γ be a rectangular dimer model. Then, ∆Γ is a parallelogram by Proposition 7.11(2).
Let e1, e2, e3, e4 be the edges of ∆Γ ordered cyclically in the anti-clockwise direction.
In particular, {e1, e3} and {e2, e4} are pairs of edges that are parallel. We may assume
that the slopes of z1, . . . , zr are the outer normal vector of e1, in which case the zigzag
paths having the slope −[zi] correspond to e3. Then, in a similar way as above, we have
the zigzag paths x1, . . . , xs (resp. y1, . . . , yt) such that [xj ] (resp. [yk]) is the outer normal
vector of e2 (resp. e4).
In both cases, we see that any pair of zigzag paths in x1, . . . , xs (resp. y1, . . . , yt) does not have
intersections on the universal cover by Lemma 3.15 because Γ is isoradial.
Next, we consider νzigX (Γ, {z1, . . . , zr}) for the case of r ̸= 1 (see Remark 6.4 for the case of
r = 1). By Lemma 7.7 and the fact that there is no intersection between x1, . . . , xs, we see that
the edges removed by (zig-5) are bypasses added in (zig-4). Furthermore, by Observations 7.3
and 7.4 any zigzag path passing through a bypass takes the form x̃′j . Since the slopes of x1, . . . , xs
are all the same in our situation, those of x′1, . . . , x
′
s are all the same (see Proposition 7.9). Thus,
any bypass on the universal cover of νzigX (Γ, {z1, . . . , zr}) is either
(i) a self-intersection of a zigzag path x̃′j , or
(ii) the intersection of a pair of zigzag paths x̃′j , x̃
′
j′ with [x′j ] = [x′j′ ].
We also see that an edge which is either (i) or (ii) is certainly a bypass, because of Observation 7.4
and the fact that such an intersection can not appear in the irrelevant part. Moreover, if
a bypass is the intersection of zigzag paths x̃′j and x̃′j′ , then they have another intersection
because [xj ] = [xj′ ], and such an intersection is also a bypass. Thus, if there exists a bypass that
can not be removed by (zig-5), the consistency condition is prevented. Therefore, we can remove
all bypasses added in (zig-4) by using (zig-5), and hence we may skip these operations. ■
42 A. Higashitani and Y. Nakajima
8 Combinatorial mutations of the PM polygon are realized
by extended deformations
Throughout this section, we still keep the notation of Sections 4 and 6 unless otherwise stated.
In this section, we show that the combinatorial mutation of the PM polygon of a consistent
dimer model coincides with the PM polygon of the deformed dimer model (see Theorem 8.3).
First, we observe the relationship between the deformation data (see Definition 4.1) and the
mutation data (see Definition 5.1).
Setting 8.1. Let Γ be a reduced consistent dimer model, and ∆Γ be the PM polygon of Γ.
We take a type I zigzag path z of Γ with v := [z] ∈ Z2. Then, by Proposition 3.5 there is an
edge E of ∆Γ whose outer normal vector is v. Since ∆Γ is determined up to translation, there
is ambiguity concerning the position of the origin. Thus, we fix the origin 0 for ∆Γ so that
0 ∈ ∆Γ. Let w := −v, and consider
hmax = hmax(∆Γ, w) := max{⟨w, u⟩ |u ∈ ∆Γ},
hmin = hmin(∆Γ, w) := min{⟨w, u⟩ |u ∈ ∆Γ}.
Now, we let r := −hmin and assume that r ≤
∣∣ZI
v(Γ)
∣∣. Since the length of the line segments of E
is |E ∩N | − 1 and this is equal to |Zv(Γ)| by Proposition 3.5, we have
−hmin = r ≤ |ZI
v(Γ)| ≤ |Zv(Γ)| = |E ∩N | − 1.
Thus ∆Γ admits the combinatorial mutation with respect to w (see Remark 5.3). Let ℓ(z) := 2n.
Then, by Lemma 3.9 we have
n = ℓ(z)/2 = |P ∩ z|+ ⟨h(P,Pi), w⟩ = |P ∩ z|+ ⟨h(P,P0)− h(Pi,P0), w⟩
= |P ∩ z|+ ⟨h(P,P0), w⟩ − ⟨h(Pi,P0), w⟩,
where P is a perfect matching on Γ, P0 is the reference perfect matching, and Pi ∈ PMmax(z).
Since h(Pi,P0) is a lattice point on E by Lemma 3.8, we have ⟨h(Pi,P0), w⟩ = hmin. If P ∈
PMmin(z), then |P∩ z| = 0 by Lemma 3.12, and this means that ⟨h(P,P0), w⟩ = hmax. Thus, we
have n = hmax − hmin = width(∆Γ, w).
We show how the correspondence between mutation data and deformation data in Table 8.1.
Mutation data Deformation data
w −v
hmin −r
hmax h
width(∆Γ, w) n
Table 8.1. Relationships between the mutation data and the deformation data.
Using the integers r, h, we take type I zigzag paths z1, . . . , zr and the zig-deformation
(resp. zag-deformation) parameter X (resp. Y) with respect to z1, . . . , zr as in Setting 6.1. We
then have the deformed consistent dimer models νzigX (Γ) = νzigX (Γ, {z1, . . . , zr}) and νzagY (Γ) =
νzagY (Γ, {z1, . . . , zr}).
We determine the origin of the PM polygons ∆
νzigX (Γ)
and ∆νzagY (Γ) as follows. First, there are
zigzag paths y′1, . . . , y
′
t on νzigX (Γ) whose slope respectively corresponds to the one of zigzag paths
y1, . . . , yt on Γ by Proposition 7.6. Then, we put ∆
νzigX (Γ)
on ∆Γ so that the edges corresponding
to y′1, . . . , y
′
t respectively coincide with the edges of ∆Γ corresponding to y1, . . . , yt. We determine
Deformations of Dimer Models 43
the origin for ∆
νzigX (Γ)
so that it is in the same position as the origin for ∆Γ. Considering the
zigzag paths on νzagY (Γ) obtained from the zigzag paths x1, . . . , xs on Γ, we can also determine
the origin for ∆νzagY (Γ).
Remark 8.2. In Setting 8.1, we assumed that r ≤ |ZI
v(Γ)| for defining the deformation data.
As we mentioned in Remark 4.2, even if the number of type I zigzag paths is insufficient, we
can sometimes change a type II zigzag path into a type I zigzag path without changing the PM
polygon by using mutations of dimer models (see Appendix A). Moreover, it is known that for
a given lattice polygon P there exists an isoradial dimer model giving P as the PM polygon
by [14], in which case all zigzag paths are type I (see Definition 3.4), and hence
∣∣ZI
v(Γ)
∣∣ =
|Zv(Γ)|. Thus, if −hmin ≤ |E ∩N | − 1 we can find a certain isoradial dimer model Γ satisfying
−hmin = r ≤ |ZI
v(Γ)| = |E ∩N | − 1.
For any edge E of ∆Γ as in Setting 8.1, we take a primitive lattice element uE ∈ N such
that ⟨w, uE⟩ = 0. Here, there are two choices of uE and we fix uE as follows. We recall the
primitive lattice element h(P′
z,Pz) given in Settings 7.8, which satisfies ⟨[z], h(P′
z,Pz)⟩ = 0.
We let uE := h(P′
z,Pz), and hence ⟨w, uE⟩ = ⟨−[z], uE⟩ = 0. We set the line segment F as
F := conv{0, uE}. Using this with Setting 8.1 (and also Table 8.1), our main theorem can be
stated as follows.
Theorem 8.3. Let Γ be a reduced consistent dimer model with 0 ∈ ∆Γ. Then, we have
mutw(∆Γ, F ) = ∆
νzigX (Γ,{z1,...,zr}),
mutw(∆Γ,−F ) = ∆νzagY (Γ,{z1,...,zr}).
Proof. We will prove the first equation. The other one follows from a similar argument.
First, we show that
φ(∆∗
Γ) = ∆∗
νzigX (Γ,{z1,...,zr})
,
where φ = φw,F is the map given in (5.2).
Let E1 := E,E2, . . . , Em be the edges of ∆Γ ordered cyclically in the anti-clockwise direction.
As we mentioned in Setting 8.1, we suppose that 0 ∈ ∆Γ. Let w1 := w,w2, . . . , wm be inner
normal vectors corresponding to E1, . . . , Em, respectively (see Figure 8.1). Also, we let vi = −wi
for i = 1, . . . , r, which are the outer normal vectors corresponding to Ei. We then consider
u ∈ ∆Γ such that ⟨w1, u⟩ = hmax(∆Γ, w1) = h; that is, we consider whmax(∆Γ), which is either
a vertex or an edge of ∆Γ. If whmax(∆Γ) is an edge, we easily see that it is parallel to E, in which
case we may write Ea := whmax(∆Γ) for some 1 < a < m. If whmax(∆Γ) is a vertex, we set the
edges intersecting at whmax(∆Γ) as Ea−1, Ea+1 and set Ea = ∅ where 1 < a < m. Recall that
by Proposition 3.5, for each Ei ̸= ∅ there exist zigzag paths on Γ such that the slopes coincide
with vi, and the set of such zigzag paths is denoted by Zvi = Zvi(Γ).
First, we consider the edge E1 and zigzag paths in Zv1 = Z(−w). By definition, we have
⟨−v1, uE⟩ = 0 and {z1, . . . , zr} ⊆ ZI
(−w) ⊆ Z(−w). If |Z(−w)| > r, then there exists a zigzag path
in Z(−w) that is not in {z1, . . . , zr}. If Ea ̸= ∅, we have zigzag paths in Zva . Since E1 and Ea
are parallel, v1 and va are linearly dependent, and hence ⟨va, uE⟩ = 0. Then, we see that zigzag
paths in Zva do not intersect with any type I zigzag path z satisfying [z] = v1 in the universal
cover (see Lemma 3.15).
Next, we consider the edges E2, . . . , Ea−1 of ∆Γ and zigzag paths in Z1 := Zv2 ∪ · · · ∪ Zva−1 .
We see that vi with i = 2, . . . , a−1 satisfies ⟨−vi, uE⟩ < 0 by a choice of the edges E2, . . . , Ea−1.
By the same argument as in the proof of Proposition 7.1, we see that the zigzag paths in Z1
intersect with a type I zigzag path z satisfying [z] = v1 precisely once in the universal cover (see
44 A. Higashitani and Y. Nakajima
wm
w1 = ww2
wa−1 wa
wa+1
uE
0
Figure 8.1. The PM polygon ∆Γ and its inner normal vectors (the case where the origin is contained
in the strict interior of ∆Γ).
Lemma 3.15); specifically, they intersect with z in Zag(z). Thus, we have Z1 = {x1, . . . , xs},
and for each i = 2, . . . , a− 1 the vector vi satisfies vi = [xj ] for some j = 1, . . . , s.
Next, we consider the edges Ea+1, . . . , Em of ∆Γ and zigzag paths in Z2 := Zva+1 ∪ · · ·
∪ Zvm . We see that vi with i = a + 1, . . . ,m satisfies ⟨−vi, uE⟩ > 0 by the choice of the edges
Ea+1, . . . , Em. By a similar argument as above, we see that the zigzag paths in Z2 intersect
with a type I zigzag path z satisfying [z] = v1 precisely once in the universal cover; specifically,
they intersect with z in Zig(z). Thus, we have Z2 = {y1, . . . , yt}, and for each i = a+ 1, . . . ,m
the vector vi satisfies vi = [yk] for some k = 1, . . . , t.
Collectively, we see that any zigzag path of Γ takes one of the following forms:
� z1, . . . , zr,
� z′1, . . . , z
′
r′ contained in Z(−w)\{z1, . . . , zr} for w = −v1 with ⟨w, uE⟩ = 0 if |Z(−w)| > r,
� z′′1 , . . . , z
′′
r′′ contained in Zw for w = −v1 with ⟨w, uE⟩ = 0 if Ea ̸= ∅,
� xj where j = 1, . . . , s, in which case it satisfies ⟨−[xj ], uE⟩ < 0,
� yk where k = 1, . . . , t, in which case it satisfies ⟨−[yk], uE⟩ > 0.
The slopes of these zigzag paths give the supporting hyperplanes of ∆Γ by Proposition 3.5. More
precisely, if ∆Γ contains the origin 0 as an interior lattice point, then
H−[ζ],≥−kζ = {u ∈ NR | ⟨−[ζ], u⟩ ≥ −kζ}
is the supporting hyperplane of ∆Γ for any zigzag path ζ of Γ and a certain positive integer kζ .
If the origin 0 lies on the boundary of ∆Γ, kζ is replaced by 0 for the zigzag paths corresponding
to the edges that contain 0. By Proposition 5.6 and its proof, ∆∗
Γ can be written as ∆∗
Γ =
Q + C, where Q is a polygon and C is a polyhedral cone. Since the set of the slopes of
zigzag paths of Γ coincides with {v1, . . . , vm} if we identify the same slopes, we see that the
set {u1, . . . , up, u′1, . . . , u′q}, which generates Q and C in the proof of Proposition 5.6, is given
by {w1, . . . , wm} in our situation. In what follows, we assume that 0 is contained in the strict
interior of ∆Γ, in which case ∆∗
Γ = Q and Q = conv
({
1
k1
w1, . . . ,
1
km
wm
})
for some positive
integers ki, giving the supporting hyperplanes Hwi,≥−ki of ∆Γ for i = 1, . . . ,m. We remark that
1
ka
wa appears in the above generating set if Ea ̸= ∅. Since ⟨wi, uE⟩ ≥ 0 for i = 1, a, a+1, . . . ,m
and ⟨wi, uE⟩ < 0 for i = 2, . . . , a− 1, we see that
φ(∆∗
Γ) = conv
({
1
k1
w1,
1
k2
w′
2, . . . ,
1
ka−1
w′
a−1,
1
ka
wa (if Ea ̸= ∅),
1
ka+1
wa+1, . . . ,
1
km
wm
})
, (8.1)
where w′
i := wi − ⟨wi, uE⟩w for i = 2, . . . , a − 1. We also note that when Ea = ∅, we can take
the positive integer ka so that the line {u ∈ NR | ⟨v1, u⟩ = −ka}, which is parallel to E1, passes
Deformations of Dimer Models 45
through the vertex of ∆Γ that is the intersection of Ea−1 and Ea+1. By the choice of ka, we have
⟨ 1
ka
v1, u⟩ ≥ −1 for any u ∈ ∆Γ; thus
1
ka
v1 =
1
ka
(−w1) ∈ ∆∗
Γ and hence 1
ka
v1 =
1
ka
(−w1) ∈ φ(∆∗
Γ).
We then consider the deformed dimer model νzigX (Γ) = νzigX (Γ, {z1, . . . , zr}). By Observa-
tion 7.2, the lift of any zigzag path of the form z′i or z′′i on the universal cover is contained in
some irrelevant part, and hence it does not change even if we apply the extended deformation.
Also, by Proposition 7.9, we have the zigzag paths x′1, . . . , x
′
s on νzigX (Γ) satisfying
−[x′j ] = −[xj ]− ⟨[xj ], h(P′
z,Pz)⟩[z] = wi − ⟨wi, uE⟩w
for j = 1, . . . , s and some i = 2, . . . , a− 1. Furthermore, by Proposition 7.6, we have the zigzag
paths y′1, . . . , y
′
t on νzigX (Γ) satisfying
−[y′k] = −[yk] = wi
for k = 1, . . . , t and some i = a+1, . . . ,m. Thus, we see that the zigzag paths xj , yk vary as they
satisfy the condition (5.2) when we apply the extended deformation νzigX to Γ. In addition, we
have the zigzag path of the form zi,j defined in (zig-3). Thus, the zigzag paths on the consistent
dimer model νzigX (Γ) are
{z′i}1≤i≤r′ (if |Z(−w)| > r), {z′′i }1≤i≤r′′ (if Ea ̸= ∅),
{x′j}1≤j≤s, {y′k}1≤k≤t, and {zi,j} 1≤i≤r
1≤j≤pi
.
By the description of their slopes and Proposition 3.5, we see that the inner normal vectors
of ∆
νzigX (Γ)
are
{w1, w
′
2, . . . , w
′
a−1, wa = −w1, wa+1, . . . , wm},
and these vectors give the supporting hyperplanes of ∆
νzigX (Γ)
just like for ∆Γ. Here, w1 appears
in the above set if |Z(−w1)| = |Zv1 | > r, but we always have 1
k1
w1 ∈ ∆∗
νzigX (Γ)
by the same
argument as we used for showing 1
ka
(−w1) ∈ ∆∗
Γ above, whereas wa certainly appears since
[zi,j ] = −[zi] = −w1 = wa (see Lemma 7.5). Thus, we have
∆∗
νzigX (Γ)
= conv
({
1
k1
w1,
1
k2
w′
2, . . . ,
1
ka−1
w′
a−1,
1
ka
wa,
1
ka+1
wa+1, . . . ,
1
km
wm
})
.
By the description (8.1) and the fact that 1
k1
w1 and 1
ka
wa = 1
ka
(−w1) are contained in both
φ(∆∗
Γ) and ∆∗
νzigX (Γ)
, we see that φ(∆∗
Γ) = ∆∗
νzigX (Γ)
.
The case where the origin 0 lies on the boundary of ∆Γ can be proved by a similar argument
if we consider the hyperplane {u ∈ NR | ⟨−[ζ], u⟩ ≥ 0} instead of {u ∈ NR | ⟨−[ζ], u⟩ ≥ −kζ} for
the zigzag paths corresponding to the edges that contain 0, in which case −[ζ] will be a generator
of a polyhedral cone C.
By Propositions 5.6(ii) and 5.7, we conclude that mutw(∆Γ, F ) = ∆
νzigX (Γ)
. ■
Since mutw(∆Γ, F ) ∼= mutw(∆Γ,−F ) (see Remark 5.3), we immediately have the following.
Corollary 8.4. We have
∆
νzigX (Γ,{z1,...,zr})
∼= ∆νzagY (Γ,{z1,...,zr}).
That is, they are GL(2,Z)-equivalent.
46 A. Higashitani and Y. Nakajima
We next show that the extended zig-deformation and zag-deformation are mutually inverse
operations on the level of the associated PM polygon as in Corollary 8.6 below. However, it is
not true on the level of dimer models as we saw in Example 4.9.
Setting 8.5. Let νzigX (Γ, {z1, . . . , zr}) be the reduced consistent dimer model. We consider the
following deformation data for νzigX (Γ, {z1, . . . , zr}). Let zi,j be a type I zigzag path which satisfies
[zi,j ] = −v =: w (see Proposition 7.5). Since {zi,j} 1≤i≤r
1≤j≤pi
is the subset of type I zigzag paths,
we have |ZI
w(ν
zig
X (Γ))| ≥
∑r
i=1 pi = h. We take a subset {z′1, . . . , z′h} of type I zigzag paths
of νzigX (Γ), and perform the same procedure as in Setting 6.1. Then we have the zag-deformation
parameter Y ′ = {Y ′
1 , . . . , Y
′
h} of the weight q′ = (q′1, . . . , q
′
h) with
∑h
i=1 q
′
i = r. We can use the
same arguments for the case of νzagY (Γ, {z1, . . . , zr}). Specifically, we take a subset {z′′1 , . . . , z′′h}
of type I zigzag paths on νzagY (Γ) and have the zig-deformation parameter X ′ = {X ′
1, . . . , X
′
h} of
the weight p′ = (p′1, . . . , p
′
h) with
∑h
i=1 p
′
i = r.
Corollary 8.6. Let the notation be the same as in Setting 8.5. Then, we have
∆
νzagY′ (ν
zig
X (Γ,{zi}ri=1),{z′ℓ}
h
ℓ=1)
= ∆Γ and ∆
νzigX′(ν
zag
Y (Γ,{zi}ri=1),{z′′ℓ }
h
ℓ=1)
= ∆Γ.
Proof. This follows from Proposition 5.5(1) and Theorem 8.3. ■
As we mentioned in Section 1, the combinatorial mutation of Fano polygons is important
from the viewpoint of mirror symmetry and the classification of Fano manifolds. To study
the combinatorial mutation of Fano polygons using extended deformations of dimer models, we
assume that the polygons ∆Γ, ∆νzigX (Γ,{z1,...,zr}) and ∆νzagY (Γ,{z1,...,zr}) contain the origin in their
strict interiors. Then, we have the following corollary.
Corollary 8.7. Let the notation be the same as above. Then, we see that ∆Γ is Fano if and
only if ∆
νzigX (Γ,{z1,...,zr}) (resp. ∆νzagY (Γ,{z1,...,zr})) is Fano.
Proof. This follows from Proposition 5.5(2) and Theorem 8.3. ■
Note that since any lattice polygon can be obtained as the PM polygon of a reduced consistent
dimer model by Theorem 2.6, we can discuss the combinatorial mutation of a polygon in terms
of the extended zig-deformations and zag-deformation by the results shown in this section.
A Mutations of dimer models
In this section, we introduce another operation, called the mutation of dimer models. From
the viewpoint of physics, dimer models and their mutations correspond to quiver gauge theories
and Seiberg duality. The mutation of dimer models can be defined for each quadrangle face of
a dimer model, and the operation called spider move (e.g., [6, 13]), which is the inverse operation
shown in Figure A.1, is the main component used in defining this mutation.
We note that there are two types of the spider move (and hence the mutation) depending on
the color of the two interior nodes.
Definition A.1 (mutation of dimer models). Let Γ be a dimer model. We pick a quadrangle
face f ∈ Γ2. Then, the mutation of Γ at f , denoted by µf (Γ), is the operation consisting of the
following procedures:
(I) If there exist black nodes on the boundary of f that are not 3-valent, we apply split moves
to those nodes and make them 3-valent as shown in Figure A.2.
Deformations of Dimer Models 47
spider move
Figure A.1.
(II) We apply the spider move to f (see Figure A.1).
(III) If the resulting dimer model contains 2-valent nodes, we remove them by applying the join
moves.
split move
f f
Figure A.2.
Applying a mutation at a quadrangle face, we obtain the new dimer model from a given one,
although the mutation sometimes induces an isomorphic one. We also remark that the mutation
is an involutive operation; that is, µf (µf (Γ)) = Γ holds. We say that dimer models Γ and Γ′ are
mutation-equivalent if they are transformed into each other by repeating the mutation of dimer
models. Moreover, we also see that the join, split and spider moves do not change the slopes of
zigzag paths and preserve conditions in Definition 3.3. Thus we have the following.
Proposition A.2. A mutation of dimer models turns consistent dimer models into consistent
dimer models associated with the same lattice polygon.
Note that it has been conjectured that all consistent dimer models associated with the same
lattice polygon are mutation-equivalent. This conjecture is still open in general (see [6, pp. 396–
397]). We note that partial answers were given in several papers (e.g., [5, 13, 15, 27]).
Remark A.3. A mutation of a dimer model is also defined as the dual of the mutation of
a quiver with potential (= QP) in the sense of [9] (see also [5, Section 7.2], [27, Section 4]).
Although we can consider the mutation of a QP for any vertex of the quiver having no loops and
2-cycles, the resulting QP is not necessarily the dual of a dimer model. To make the resulting
quiver the dual of a dimer model, we need to assume that the mutated vertex has two incoming
(equivalently, two outgoing) arrows, which is equivalent to assuming that the face of a dimer
model corresponding to such a vertex is a quadrangle.
In the theory of (extended) deformations of dimer models, type I zigzag paths are important.
We can use mutations to change a type II zigzag path into type I as in the example below.
Example A.4. We consider the following dimer model Γ (the image on the left). Since the
face 0 is a quadrangle, we can apply the mutation and obtain the dimer model µ0(Γ) as follows.
48 A. Higashitani and Y. Nakajima
0
0µ0
We can see that the zigzag path on Γ whose slope is (−1, 1) or (1,−1) is type II. On the
other hand, we see that µ0(Γ) is isoradial, and hence all zigzag paths are type I.
B Large examples
As we mentioned in Remark 6.4 and Proposition 7.12, we sometimes skip the operations (zig-4)
and (zig-5) (resp. (zag-4) and (zag-5)) when we define the extended deformation νzigX (Γ, {z1, . . . ,
zr}) (resp. νzagY (Γ, {z1, . . . , zr})) of a consistent dimer model Γ. However, as the following ex-
ample shows, (zig-4) and (zig-5) (resp. (zag-4) and (zag-5)) are indispensable to define extended
deformations compatible with combinatorial mutations of polygons, as shown in Theorem 8.3.
For example, we often encounter such a situation when we consider a consistent dimer model
whose PM polygon is relatively large.
Example B.1. We consider the lattice polygon P shown on the left-hand side of Figure B.1. We
assume that the double circle stands for the origin 0. We consider the edge E whose primitive
inner normal vector is w = (0,−1) with hmin = −3 and hmax = 1. We take uE = (−1, 0), which
satisfies ⟨w, uE⟩ = 0, and consider the line segment F = conv{0, uE}. Then, the combinatorial
mutation mutw(P, F ) of the polygon P is as shown on the right-hand side of Figure B.1.
mutw(−, F )
Figure B.1. The lattice polygons P and mutw(P, F ) for w = (0,−1) and F = conv{0, (−1, 0)}.
Next, we consider the dimer model Γ shown in Figure B.2. The zigzag paths on this dimer
model Γ are depicted in Figure B.3. One can check that Γ is consistent, and hence the PM
polygon ∆Γ coincides with P by Proposition 3.5.
We now consider the extended zig-deformation of Γ that realizes mutw(P, F ) as the PM
polygon (see Theorem 8.3). To do this, we first fix the deformation data (see Definition 4.1) as
follows. First, the zigzag path zi on Γ is type I with ℓ(zi) = 8, and its slope is [zi] = (0, 1) = −w
for i = 1, . . . , 4. Let r := −hmin = 3 and h := hmax = 1 (see Table 8.1). These satisfy r = 3 <∣∣ZI
(−w)(Γ)
∣∣ = 4 and r + h = ℓ(zi)/2 = 4. We take the set of type I zigzag paths {z1, z2, z3}, and
consider the extended zig-deformation νzigX (Γ, {z1, z2, z3}) of Γ at {z1, z2, z3} with respect to X ,
where X is the zig-deformation parameter (see Setting 6.1) defined as follows. To define X , we
focus on x1, x2, x3 shown in Figure B.3, which are the zigzag paths intersecting with zi at some
zags of zi. They satisfy m1 := |x1 ∩ zi| = 1, m2 := |x2 ∩ zi| = 1, and m3 := |x3 ∩ zi| = 2 for
any i; thus we consider the set of sub-zigzag paths
{
x1 = x
(1)
1 , x2 = x
(1)
2 , x
(1)
3 , x
(2)
3
}
. Here, we
fix the intersection of z3 and x3 marked by ✓ in Figure B.3 as the starting edge of x
(1)
3 . We
Deformations of Dimer Models 49
Figure B.2. A consistent dimer model Γ whose PM polygon coincides with P .
z1 z2 z3 z4 z′1 z′2 z′3 z′4
y1
y2
y3
y4
x1
x2
x3
✓
Figure B.3. The zigzag paths of Γ.
then set the zig-deformation parameter X := {X1, X2, X3} with respect to {z1, z2, z3}, where
X1 =
{
x1 ∩ z1 = x
(1)
1 ∩ z1
}
, X2 =
{
x2 ∩ z2 = x
(1)
2 ∩ z2
}
and X3 =
{
x
(1)
3 ∩ z3, x
(2)
3 ∩ z3
}
, and
hence the weight of X is p = (0, 0, 1).
Using these deformation data, we apply the operations (zig-1)–(zig-3) to Γ. This gives us
the dimer model shown in the left of Figure B.4. Here, we can easily see that the slopes of
zigzag paths on this dimer model do not correspond bijectively to the primitive side segments of
mutw(P, F ), and thus we can not obtain Theorem 8.3 without the operations (zig-4) and (zig-5).
We therefore apply (zig-4) – that is, we insert some bypasses. Then we have the dimer model
νzigX (Γ) = νzigX (Γ, {z1, z2, z3}), as shown in the right of Figure B.4. Some zigzag paths on νzigX (Γ)
are given in Figure B.5. For example, the zigzag path x3 on Γ behaves as Figure 7.7 in
{
x
(1)
3 ∩zi,
x
(2)
3 ∩zi | i = 1, 2
}
and behaves as Figure 7.8 in X3 =
{
x
(1)
3 ∩z3, x
(2)
3 ∩z3
}
by our choice of X , and
hence we obtain the zigzag path x′3 on νzigX (Γ) as in Figure B.5. By Proposition 7.1, this dimer
model is non-degenerate, but it is not consistent. Indeed, we can see that some nodes appearing
on the zigzag path x′3 shown in Figure B.5 are not properly ordered (see Definition 3.3(4)).
By Lemma 7.7, some edges constituting the zigzag paths x′1, x
′
2, x
′
3 shown in Figure B.5
might be removed by the operation (zig-5). Thus, paying attention to these zigzag paths, we
apply (zig-5) to νzigX (Γ) and make it consistent. Namely, we remove pairs of edges (1)–(5)
shown in the type A (left) of Figure B.6, which are the intersections of pairs of zigzag paths
50 A. Higashitani and Y. Nakajima
Figure B.4. The dimer model obtained by applying (zig-1)–(zig-3) to Γ (left), and the dimer model
νzigX (Γ, {z1, z2, z3}) obtained by applying (zig-1)–(zig-4) to Γ (right).
x′1
x′2
x′3
Figure B.5. The zigzag paths x′
1, x
′
2, x
′
3 of νzigX (Γ, {z1, z2, z3}).
on the universal cover that intersect with each other in the same direction more than once,
from νzigX (Γ) with this order. Then we have the dimer model shown in the left of Figure B.7.
Applying (join), we finally have the dimer model νzigX (Γ, {z1, z2, z3}) shown in the right of Figu-
re B.7.
There are also other ways to remove edges. For example, if we remove pairs of edges (1)–(5)
shown in the type B (right) of Figure B.6, then we have the dimer model shown in the left of
Figure B.8, and applying (join) we have the dimer model νzigX (Γ, {z1, z2, z3}) shown in the right
of Figure B.8.
(1)
(1)
(2)
(2)
(3) (3)
(4)
(4)
(5)
(5)
(1)
(1)
(2)
(2)
(3)
(3)
(4)
(4)
(5)
(5)
Type A Type B
Figure B.6. The two ways to remove edges from νzigX (Γ, {z1, z2, z3}) by (zig-5).
Let ΓA (resp. ΓB) be the deformed dimer model shown in the right of Figure B.7 (resp. Fi-
gure B.8). We can check that ΓA and ΓB are not isomorphic, but they are mutation-equivalent.
Indeed, by applying the mutations of ΓB at the faces 1, . . . , 10 in this order (see Figure B.9), we
recover ΓA. Here, we recall that a mutation of a dimer model can be defined at any quadrangle
Deformations of Dimer Models 51
(join)
Figure B.7. The dimer model νzigX (Γ, {z1, z2, z3}) obtained from the type A of Figure B.6.
(join)
Figure B.8. The dimer model νzigX (Γ, {z1, z2, z3}) obtained from the type B of Figure B.6.
face. Although some faces indexed by {1, . . . , 10} are not quadrangle, such faces will become
quadrangles during the process of this series of mutations.
The PM polygons of the dimer models ΓA and ΓB are the same (see Proposition A.2), and it
coincides with the lattice polygon shown on the right-hand side of Figure B.1 by Theorem 8.3.
1
2
3
4
5
6
7
8
9
10 The mutation µf
where f = 1, . . . , 10
Figure B.9. The mutations of ΓB (left) at the faces 1, . . . , 10 induce ΓA (right).
Acknowledgements
The authors would like to thank Alexander Kasprzyk for valuable lectures and discussions on the
combinatorial mutation of Fano polygons. The authors would also like to thank the anonymous
referees for their numerous valuable comments and suggestions. The first author is supported
by JSPS Grant-in-Aid for Scientific Research (C) 20K03513. The second author was supported
by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and
is supported by JSPS Grant-in-Aid for Early-Career Scientists 20K14279.
52 A. Higashitani and Y. Nakajima
References
[1] Akhtar M., Coates T., Corti A., Heuberger L., Kasprzyk A., Oneto A., Petracci A., Prince T., Tveiten K.,
Mirror symmetry and the classification of orbifold del Pezzo surfaces, Proc. Amer. Math. Soc. 144 (2016),
513–527, arXiv:1501.05334.
[2] Akhtar M., Coates T., Galkin S., Kasprzyk A.M., Minkowski polynomials and mutations, SIGMA 8 (2012),
094, 707 pages, arXiv:1212.1785.
[3] Beil C., Ishii A., Ueda K., Cancellativization of dimer models, arXiv:1301.5410.
[4] Bocklandt R., Consistency conditions for dimer models, Glasg. Math. J. 54 (2012), 429–447,
arXiv:1104.1592.
[5] Bocklandt R., Generating toric noncommutative crepant resolutions, J. Algebra 364 (2012), 119–147,
arXiv:1104.1597.
[6] Bocklandt R., A dimer ABC, Bull. Lond. Math. Soc. 48 (2016), 387–451, arXiv:1510.04242.
[7] Broomhead N., Dimer models and Calabi–Yau algebras, Mem. Amer. Math. Soc. 215 (2012), viii+86 pages,
arXiv:0901.4662.
[8] Coates T., Corti A., Galkin S., Golyshev V., Kasprzyk A., Mirror Symmetry and Fano Manifolds, in
European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, 285–300, arXiv:1212.1722.
[9] Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations. I. Mutations,
Selecta Math. (N.S.) 14 (2008), 59–119, arXiv:0704.0649.
[10] Duffin R.J., Potential theory on a rhombic lattice, J. Combinatorial Theory 5 (1968), 258–272.
[11] Franco S., Hanany A., Vegh D., Wecht B., Kennaway K.D., Brane dimers and quiver gauge theories, J. High
Energy Phys. 2006 (2006), no. 1, 096, 48 pages, arXiv:hep-th/0504110.
[12] Galkin S., Usnich A., Mutations of potentials, Preprint IPMU 10–0100, 2010.
[13] Goncharov A.B., Kenyon R., Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46
(2013), 747–813, arXiv:1107.5588.
[14] Gulotta D.R., Properly ordered dimers, R-charges, and an efficient inverse algorithm, J. High Energy Phys.
2008 (2008), no. 10, 014, 31 pages, arXiv:0807.3012.
[15] Hanany A., Seong R.K., Brane tilings and reflexive polygons, Fortschr. Phys. 60 (2012), 695–803,
arXiv:1201.2614.
[16] Hanany A., Vegh D., Quivers, tilings, branes and rhombi, J. High Energy Phys. 2007 (2007), no. 10, 029,
35 pages, arXiv:hep-th/0511063.
[17] Higashitani A., Nakajima Y., Combinatorial mutations of Newton–Okounkov polytopes arising from plabic
graphs, Adv. Stud. Pure Math., to appear, arXiv:2107.04264.
[18] Ilten N.O., Mutations of Laurent polynomials and flat families with toric fibers, SIGMA 8 (2012), 047,
7 pages, arXiv:1205.4664.
[19] Ishii A., Ueda K., A note on consistency conditions on dimer models, in Higher Dimensional Algebraic
Geometry, RIMS Kôkyûroku Bessatsu, Vol. B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 143–164,
arXiv:1012.5449.
[20] Ishii A., Ueda K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 3405–3466,
arXiv:0905.0059.
[21] Iyama O., Nakajima Y., On steady non-commutative crepant resolutions, J. Noncommut. Geom. 12 (2018),
457–471, arXiv:1509.09031.
[22] Kasprzyk A., Nill B., Prince T., Minimality and mutation-equivalence of polygons, Forum Math. Sigma 5
(2017), e18, 48 pages, arXiv:1501.05335.
[23] Kennaway K.D., Brane tilings, Internat. J. Modern Phys. A 22 (2007), 2977–3038, arXiv:0706.1660.
[24] Kenyon R., An introduction to the dimer model, in School and Conference on Probability Theory, ICTP Lect.
Notes, Vol. 17, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 267–304, arXiv:math.CO/0310326.
[25] Kenyon R., Schlenker J.M., Rhombic embeddings of planar quad-graphs, Trans. Amer. Math. Soc. 357
(2005), 3443–3458, arXiv:math-ph/0305057.
[26] Mercat C., Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), 177–216,
arXiv:0909.3600.
https://doi.org/10.1090/proc/12876
https://arxiv.org/abs/1501.05334
https://doi.org/10.3842/SIGMA.2012.094
https://arxiv.org/abs/1212.1785
https://arxiv.org/abs/1301.5410
https://doi.org/10.1017/S0017089512000080
https://arxiv.org/abs/1104.1592
https://doi.org/10.1016/j.jalgebra.2012.03.040
https://arxiv.org/abs/1104.1597
https://doi.org/10.1112/blms/bdv101
https://arxiv.org/abs/1510.04242
https://doi.org/10.1090/S0065-9266-2011-00617-9
https://arxiv.org/abs/0901.4662
https://doi.org/10.4171/120-1/16
https://arxiv.org/abs/1212.1722
https://doi.org/10.1007/s00029-008-0057-9
https://arxiv.org/abs/0704.0649
https://doi.org/10.1016/S0021-9800(68)80072-9
https://doi.org/10.1088/1126-6708/2006/01/096
https://doi.org/10.1088/1126-6708/2006/01/096
https://arxiv.org/abs/hep-th/0504110
https://doi.org/10.24033/asens.2201
https://arxiv.org/abs/1107.5588
https://doi.org/10.1088/1126-6708/2008/10/014
https://arxiv.org/abs/0807.3012
https://doi.org/10.1002/prop.201200008
https://arxiv.org/abs/1201.2614
https://doi.org/10.1088/1126-6708/2007/10/029
https://arxiv.org/abs/hep-th/0511063
https://arxiv.org/abs/2107.04264
https://doi.org/10.3842/SIGMA.2012.047
https://arxiv.org/abs/1205.4664
https://arxiv.org/abs/1012.5449
https://doi.org/10.2140/gt.2015.19.3405
https://arxiv.org/abs/0905.0059
https://doi.org/10.4171/JNCG/283
https://arxiv.org/abs/1509.09031
https://doi.org/10.1017/fms.2017.10
https://arxiv.org/abs/1501.05335
https://doi.org/10.1142/S0217751X07036877
https://arxiv.org/abs/0706.1660
https://arxiv.org/abs/math.CO/0310326
https://doi.org/10.1090/S0002-9947-04-03545-7
https://arxiv.org/abs/math-ph/0305057
https://doi.org/10.1007/s002200000348
https://arxiv.org/abs/0909.3600
Deformations of Dimer Models 53
[27] Nakajima Y., Mutations of splitting maximal modifying modules: the case of reflexive polygons, Int. Math.
Res. Not. 2019 (2019), 470–550, arXiv:1601.05203.
[28] Nakajima Y., Semi-steady non-commutative crepant resolutions via regular dimer models, Algebr. Comb. 2
(2019), 173–195, arXiv:1608.05162.
[29] Rietsch K., Williams L., Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians,
Duke Math. J. 168 (2019), 3437–3527, arXiv:1712.00447.
[30] Schrijver A., Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics,
John Wiley & Sons, Ltd., Chichester, 1986.
[31] Ueda K., Yamazaki M., A note on dimer models and McKay quivers, Comm. Math. Phys. 301 (2011),
723–747, arXiv:math.AG/0605780.
https://doi.org/10.1093/imrn/rnx114
https://doi.org/10.1093/imrn/rnx114
https://arxiv.org/abs/1601.05203
https://doi.org/10.5802/alco.39
https://arxiv.org/abs/1608.05162
https://doi.org/10.1215/00127094-2019-0028
https://arxiv.org/abs/1712.00447
https://doi.org/10.1007/s00220-010-1101-0
https://arxiv.org/abs/math.AG/0605780
1 Introduction
1.1 Background and motivations
1.2 Results
2 Dimer models and perfect matching polygons
2.1 What is a dimer model?
2.2 Perfect matchings and the perfect matching polygon
3 Zigzag paths and their properties
3.1 Consistency conditions
3.2 Relationships between perfect matchings and zigzag paths
4 Deformations of dimer models
4.1 Definition of deformations of dimer models
4.2 Examples of deformations of dimer models
5 Combinatorial mutations of perfect matching polygons
5.1 Preliminaries on combinatorial mutations of polytopes
5.2 The perfect matching polygons of deformed dimer models
6 Extended deformations of dimer models
7 Foundations of extended deformations of dimer models
7.1 The proof of the non-degeneracy
7.2 Behaviors of zigzag paths after extended deformations
7.3 Properties of zigzag paths on deformed dimer models
7.4 Remarks on the extended deformations of hexagonal and rectangular dimer models
8 Combinatorial mutations of the PM polygon are realized by extended deformations
A Mutations of dimer models
B Large examples
References
|
| id | nasplib_isofts_kiev_ua-123456789-211638 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T11:15:08Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Higashitani, Akihiro Nakajima, Yusuke 2026-01-07T13:43:11Z 2022 Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages 1815-0659 2020 Mathematics Subject Classification: 52B20; 14M25; 14J33 arXiv:1903.01636 https://nasplib.isofts.kiev.ua/handle/123456789/211638 https://doi.org/10.3842/SIGMA.2022.030 The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon , there exists a dimer model having as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons. The authors would like to thank Alexander Kasprzyk for valuable lectures and discussions on the combinatorial mutation of Fano polygons. The authors would also like to thank the anonymous referees for their numerous valuable comments and suggestions. The first author is supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03513. The second author was supported by the World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and is supported by JSPS Grant-in-Aid for Early-Career Scientists 20K14279. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations of Dimer Models Article published earlier |
| spellingShingle | Deformations of Dimer Models Higashitani, Akihiro Nakajima, Yusuke |
| title | Deformations of Dimer Models |
| title_full | Deformations of Dimer Models |
| title_fullStr | Deformations of Dimer Models |
| title_full_unstemmed | Deformations of Dimer Models |
| title_short | Deformations of Dimer Models |
| title_sort | deformations of dimer models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211638 |
| work_keys_str_mv | AT higashitaniakihiro deformationsofdimermodels AT nakajimayusuke deformationsofdimermodels |