Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 2. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to co...
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| citation_txt | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. Boris Bychkov, Anton Kazakov and Dmitry Talalaev. SIGMA 17 (2021), 035, 30 pages |
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| description | We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 2. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter n. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of = 2 multivariate Tutte polynomial. We extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 035, 30 pages
Functional Relations on Anisotropic Potts Models:
from Biggs Formula to the Tetrahedron Equation
Boris BYCHKOV ab, Anton KAZAKOV abc and Dmitry TALALAEV abc
a) Faculty of Mathematics, National Research University Higher School of Economics,
Usacheva 6, 119048, Moscow, Russia
E-mail: bbychkov@hse.ru, dtalalaev@yandex.ru, anton.kazakov.4@mail.ru
b) Centre of Integrable Systems, P.G. Demidov Yaroslavl State University,
Sovetskaya 14, 150003, Yaroslavl, Russia
c) Faculty of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia
Received July 06, 2020, in final form March 26, 2021; Published online April 07, 2021
https://doi.org/10.3842/SIGMA.2021.035
Abstract. We explore several types of functional relations on the family of multivariate
Tutte polynomials: the Biggs formula and the star-triangle (Y −∆) transformation at the
critical point n = 2. We deduce the theorem of Matiyasevich and its inverse from the
Biggs formula, and we apply this relation to construct the recursion on the parameter n.
We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the
star-triangle transformation in the case of n = 2 multivariate Tutte polynomial, we extend
the latter to the case of valency 2 points and show that the Biggs formula and the star-
triangle transformation commute.
Key words: tetrahedron equation; local Yang–Baxter equation; Biggs formula; Potts model;
Ising model
2020 Mathematics Subject Classification: 82B20; 16T25; 05C31
1 Introduction
The theory of polynomial invariants of graphs in its current state uses many methods and
tools of integrable statistical mechanics. This phenomenon demonstrates the inherent intrusion
of mathematical physics methods into topology and combinatorics. In this paper, the main
subject of research is functional relations in the family of polynomial invariants for framed
graphs, in particular for multivariate Tutte polynomials [22], their specializations for Potts
models, multivariate chromatic and flow polynomials.
The flow generating function is closely related to the problems of electrical networks on
a graph over a finite field. Each flow defines a discrete harmonic function, and non-zero flows
can be interpreted as harmonic functions with a completely non-zero gradient. Specifically,
we discuss the full flow polynomial which is a linearization of the flow polynomial and, in par-
ticular, corresponds to the point of the compactification of the parameter space for the Biggs
model.
One of the central tools of the paper is the Biggs formula (Lemma 2.13), which connects
n-Potts models for different parameter values as a convolution with some weight over all edge
subgraphs. In particular, we offer a new proof of the theorem of Matiyasevich 2.18 about
the connection of a flow and chromatic polynomial, as a special case of the Biggs formula.
This interpretation allows us to construct an inverse statement of the theorem of Matiyasevich.
Moreover, using the connection between the flow and the complete flow polynomial, we obtain
a shift of parameters in the Potts models (Theorem 2.25).
mailto:bbychkov@hse.ru
mailto:dtalalaev@yandex.ru
mailto:anton.kazakov.4@mail.ru
https://doi.org/10.3842/SIGMA.2021.035
2 B. Bychkov, A. Kazakov and D. Talalaev
The fundamental type of correspondences on the space of the aforementioned invariants is the
star-triangle type relations (also known as “wye-delta” relations) and the associated deletion-
contraction relations. In a sense, the kinship between these relations is analogous to the role
of the tetrahedron equation in the local Yang–Baxter equation. Despite the fact that the invari-
ance of the Ising model with respect to the star-triangle transformation is very well known [2],
we have not found in the literature a full proof of the fact that the action of this transfor-
mation on the weights of an anisotropic system is a solution of the tetrahedron equation that
corresponds to the orthogonal solution of the local Yang–Baxter equation: Theorem 4.1 (parts
of this statement were mentioned in [15, 17, 21]). We offer here two new proofs of this fact.
We find them instructive due to their anticipated relation to the theory of positive orthogonal
grassmannians [14].
The identification of the Potts model and the multivariate Tutte polynomial allows us to
assert the existence of a critical point for the parameter n in the family of Tutte polynomials.
Namely, for n = 2, this model has a groupoid symmetry generated by a family of transformations
defined by the trigonometric solution of the Zamolodchikov tetrahedron equation. We extend the
star-triangle transformation for the graphs of lower valency in Section 5. In this way, we obtain
a 14-term correspondence. This extension commute with the Biggs formula. We should mention
the relation of this subject with the theory of cluster algebras. We suppose that the multivariate
Tutte polynomial on standard graphs at the critical point n = 2 corresponds to the orthogonal
version of the Lusztig variety [4] in the case of the unipotent group and the electrical variety [12]
for the symplectic group.
1.1 Organization of the paper
In Section 2, we concentrate our attention on the Biggs formalism in the Ising and Potts type
models. We define the main recurrence relations and also identify the Tutte polynomial with
the Potts model. Then, we apply the Biggs formula to the proof of theorem of Matiyasevich
and propose its inverse version. We examine in details the recursion of the Potts model with
respect to the parameter n.
In Section 3, we show that, if n = 2, then the Potts model is invariant with respect to the
star-triangle transformation given by the orthogonal solution for the local Yang–Baxter equation
and the corresponding solution for the Zamolodchikov tetrahedron equation. In Section 4 we
provide two different proofs for this fact. Both of them are interesting in the context of cluster
variables on the space of Ising models. The first proof operates with the space of boundary
measurement matrices and the second with the matrix of boundary partition function.
In Section 5, we show that the Biggs formula considered as a correspondence on the set
of multivariate Tutte polynomials commutes with the star-triangle transformation.
2 Biggs interaction models
2.1 n-Potts models and Tutte polynomial
We define the anisotropic Biggs model (interaction model) on an undirected graph G with the
set of edges E and the set of vertices V (a graph can have multiple edges and loops) as follows:
� a state σ is a map σ : V → R, where R is a commutative ring with the unit,
� the weight of the state σ is defined by the formula
WG(σ) :=
∏
e∈E
ie(δ(e)),
Functional Relations on Anisotropic Potts Models 3
where δ(e) = σ(v)−σ(w); the edge e connects the vertices v and w, the functions ie : R→ C
are even: ∀b ∈ R : ie(b) = ie(−b),
� the partition function Z(G) of a model is the following sum
Z(G) =
∑
σ
WG(σ),
where the summation is taken over all possible states σ.
Let us consider the most simple Biggs interaction models:
Definition 2.1. If R ∼= Zn and functions ie given as{
ie(0) = αe,
ie(a) = βe, ∀a 6= 0 ∈ R,
we call such model the anisotropic n-Potts model with the set of parameters αe and βe and we
denote it by M(G; ie).
In addition, if the maps ie = i do not depend on edges, then we call such model the isotropic
n-Potts model (or just n-Potts model) with parameters α and β. We denote it by M(G; i), also
we use the notation M(G;α, β).
Remark 2.2. In the case R ∼= Z2, i(0) = exp
(
J
kT
)
and i(1) = exp
(
− J
kT
)
this model can
be identified with the classic isotropic Ising model [2]. Therefore we will call any anisotropic
or isotropic 2-Potts model just Ising model.
Definition 2.3. Consider an anisotropic n-Potts model M(G; ie), we denote its partition func-
tion as Zn(G). In addition, if n = 2, we omit index 2 and write just Z(G).
Remark 2.4. For the empty graph, we define the partition function of any n-Potts model to
be equal to 1, and for a disjoint set of m points to be equal to nm.
Now we will consider the combinatorial properties of the anisotropic n-Potts models (compare
with [3, Theorem 3.2]):
Theorem 2.5. Consider an anisotropic n-Potts model M(G; i) and its partition function Zn(G).
� Let graph G be the disjoint union of graphs G1 and G2, then
Zn(G) = Zn(G1)Zn(G2).
Figure 1. The joining of two graphs by the vertex v.
� Let graph G be the joining of graphs G1 and G2 by the vertex v, then
nZn(G) = Zn(G1)Zn(G2).
4 B. Bychkov, A. Kazakov and D. Talalaev
� Consider a graph G and its edge e, where e is neither a bridge nor a loop. Consider the
graph G/e obtained by contraction of e, and the graph G\e obtained by deletion of e. Then
the following formula holds
Zn(G) = (αe − βe)Zn(G/e) + βeZn(G\e).
Proof.
1. The statement directly follows from Definition 2.1.
2. Let us rewrite the partition function Zn(G):
Zn(G) =
∑
k∈{0,...,n−1}
∑
σ : σ(v)=k
WG(σ).
Notice that i(σ(v) − σ(w)) = i(σ(v) + 1 − σ(w) − 1), therefore for any i 6= j we have the
following identity∑
σ : σ(v)=i
WG(σ) =
∑
σ : σ(v)=j
WG(σ).
Hence we obtain
Zn(G) = n
∑
σ : σ(v)=i
WG(σ), ∀ i ∈ {0, . . . , n− 1}.
Let us introduce the partial partition functions Xk :=
∑
σ : σ(v)=k
WG1(σ) and Yk :=
∑
σ : σ(v)=k
WG2(σ),
then we could rewrite
Zn(G1)Zn(G2) =
(∑
k
∑
σ : σ(v)=k
WG1(σ)
)(∑
k
∑
σ : σ(v)=k
WG2(σ)
)
= (X0 +X1 + · · ·+Xn−1)(Y0 + Y1 + . . . Yn−1) = n2X0Y0
= n(X0Y0 +X1Y1 + · · ·+Xn−1Yn−1) = n
∑
k
∑
σ : σ(v)=k
WG(σ) = nZn(G).
3. Let the edge e is neither a bridge nor a loop and denote by X the income in the partition
function of all states such that the values of the ends of e coincide and by Y another part of the
partition function (that of the distinct values of the ends of e), then
Zn(G) = αeX + βeY, Zn(G\e) = X + Y, Zn(G/e) = X
and we obtain the statement. �
Now let us recall the definition of the Tutte polynomial of a graph G.
Definition 2.6. Let us define the Tutte polynomial TG(x, y) by the deletion-contraction recur-
rence relation:
1. If an edge e is neither a bridge nor a loop, then TG(x, y) = TG\e(x, y) + TG/e(x, y).
2. If the graph G consists of i bridges and j loops, then TG(x, y) = xiyj .
Theorem 2.7 ([9, 23]). Let F (G) be a function of a graph G satisfuing the following conditions:
� F (G) = 1, if G consists of only one vertex.
Functional Relations on Anisotropic Potts Models 5
� F (G) = aF (G\e) + bF (G/e), if an edge e is not a bridge neither a loop.
� F (G) = F (G1)F (G2), if either G = G1 tG2 or the intersection G1 ∩G2 consists of only
one vertex.
Then
F (G) = ac(G)br(G)TG
(
F (K2)
b
,
F (L)
a
)
,
where K2 is a complete graph on two vertices, L is a loop, r(G) = v(G) − k(G) is a rank of G
and c(G) = e(G)−r(G) is a corank. Here and below e(G) is the number of edges in the graph G.
Now we are ready to connect the partition function Zn(G) of the isotropic n-Potts model
M(G;α, β) with the Tutte polynomial TG(x, y) of the same graph G using a well-known trick
(for instance see [3]). Let us consider the weighted partition function
Zn(G)
nk(G)
,
where k(G) is the number of connected components in the graph G. It is easy to verify that the
weighted partition function Zn(G)
nk(G) satisfies Theorem 2.7, therefore the following theorem holds:
Theorem 2.8 (Theorem 3.2 [3]). The partition function Zn(G) of the n-Potts model M(G;α, β)
coincides with the Tutte polynomial of a graph G up to a multiplicative factor
Zn(G) = nk(G)βc(G)(α− β)r(G)TG
(
α+ (n− 1)β
α− β
,
α
β
)
.
Example 2.9 (the bad coloring polynomial [9]). Consider a graph G and all possible colorings
of V (G) in n colors. Define the bad coloring polynomial as
BG(n, t) =
∑
j
bj(G,n)tj ,
here bj(G,n) is the number of colorings such that each of them has exactly j bad edges (we
call an edge “bad” if its ends have the same colors). So, easy to see that BG(n, t) = Zn(G),
here Zn(G) is the partition function of the n-Potts model M(G; t, 1). Hence, using the Theo-
rem 2.8 we immediately obtain
BG(n, t+ 1) = nk(G)tr(G)TG
(
t+ n
t
, t+ 1
)
.
2.2 n-Potts models and the theorem of Matiyasevich
The connection between the n-Potts models and Tutte polynomials allows us to give a simple
proof of the theorem of Matiyasevich about the chromatic and flow polynomials, but at first
we introduce a few definitions.
Definition 2.10. A graph A is called a spanning subgraph of a graph G, if graphs G and A
share the same set of vertices: V (G) = V (A), and the set of edges E(A) is the subset of the set
of edges E(G).
Definition 2.11. A graph A is called an edge induced subgraph (Figure 2) of a graph G, if A
is induced by a subset of the set E(G). Every edge induced subgraph A of a graph G could be
completed to the spanning subgraph A′ by adding all the vertices of G which is not contained
in the subgraph A.
6 B. Bychkov, A. Kazakov and D. Talalaev
Definition 2.12. For a n-Potts model M(G; i) we introduce the normalized partition function
as follows
Z̃n(G) =
Zn(G)
nv(G)
.
The edge induced subgraph The spanning subgraph
Figure 2. Edge induced and spanning subgraphs.
We start with the following lemma, which is a generalization of the high temperature formula
for the Ising model:
Lemma 2.13 (Biggs formula [5]). Let us consider two n-Potts models M1(G; i1) with parame-
ters α1, β1 and M2(G; i2) with parameters α2, β2. Then the normalized partition function Z1
n(G)
of the first model could be expressed in terms of the normalized partition functions of the models
of all edge induced subgraphs of the second model:
Z̃1
n(G) = qe(G)
∑
A⊆G
(
p
q
)e(A)
Z̃2
n(A),
where p = α1−β1
α2−β2 , and q = α2β1−α1β2
α2−β2
(
we assume that Z̃in(∅) = 1
)
.
Proof. Let us notice that i1 = p · i2 + q, therefore
Z̃1
n(G) =
∑
σ : V (G)→Zn
∏
e
i1(δ(e)) =
∑
σ : V (G)→Zn
∏
e
(pi2(δ(e)) + q)
=
∑
σ : V (G)→Zn
∑
A⊆G
pe(A)qe(G)−e(A)
∏
e∈E(A)
i2(δ(e)).
In order to complete the proof we consider the following term for a fixed A:∑
σ : V (G)→Zn
pe(A)qe(G)−e(A)
∏
e∈E(A)
i2(δ(e)) = qe(G)
(
p
q
)e(A) ∑
σ : V (G)→Zn
∏
e∈E(A)
i2(δ(e))
= qe(G)
(
p
q
)e(A)
nv(G)−v(A)
∑
σ : V (A)→Zn
∏
e∈E(A)
i2(δ(e))
= nv(G)qe(G)
∑
A⊆G
(
p
q
)e(A)
Z̃2
n(A). �
Proposition 2.14. Consider two anisotropic n-Potts models M1
(
G; i1e
)
and M2
(
G; i2e
)
. In the
same fashion we can obtain
Z̃1
n(G) =
∏
e∈G
qe
∑
A⊆G
∏
e∈A
pe
qe
Z̃2
n(A), (2.1)
here pe = α1
e−β1
e
α2
e−β2
e
, and qe = α2
eβ
1
e−α1
eβ
2
e
α2
e−β2
e
.
Functional Relations on Anisotropic Potts Models 7
We consider further the chromatic and flow polynomials, first of all remain some well-known
definitions.
Definition 2.15. A coloring of the set of vertices V (G) is said to be proper if the ends of each
edge have different colors.
Definition 2.16. Let G be a graph with the edge set E(G) and the vertex set V (G), let us
choose a fixed edge orientation on G. Then, a function f : E → Zn is called a nowhere-zero
n-flow if the following conditions hold:
� ∀e ∈ E(G) : f(e) 6= 0,
� ∀v ∈ V (G) :
∑
e∈M+(v)
f(e) =
∑
e∈M−(v)
f(e), where M+(v) (respectively M−(v)) is the set
of edges each of them is directed to (respectively from) v.
Next, we formulate one of the classic results of graph theory which can be found for instance
in [9]:
Theorem 2.17. The number of proper colorings of a graph G in n colors is the following
polynomial (called chromatic polynomial) in the variable n:
χG(n) = (−1)v(G)−k(G)nk(G)TG(1− n, 0).
The number of nowhere-zero n-flows of a graph G is independent on the choice of orientation
and is obtained by the following polynomial (called flow polynomial) in the variable n:
CG(n) = (−1)e(G)+v(G)+k(G)TG(0, 1− n).
Now we are ready to formulate and prove the theorem of Matiyasevich:
Theorem 2.18 (Matiyasevich [20]). Let us consider a graph G, its chromatic polynomial χG
and its flow polynomial CG, then
χG(n) =
(n− 1)e(G)
ne(G)−v(G)
∑
A⊆G
CA(n)
(1− n)e(A)
,
where the summation goes through all spanning subgraphs A.
Proof. Let us consider two n-Potts models with the special parameters: the model M1(G; i1)
with the parameters α1 = 0, β1 = 1 and the model M2(G; i2) with the parameters α2 = 1− n,
β2 = 1. By Theorem 2.8 we could express the partition function of the first model in terms
of the chromatic polynomial
χG(n) = (−1)v(G)−k(G)nk(G)TG(1− n, 0) =
(−1)v(G)−k(G)−r(G)nk(G)Z1
n(G)
nk(G)
= (−1)v(G)−k(G)−r(G)nv(G)Z̃1
n(G) = nv(G)Z̃1
n(G).
So we have
Z̃1
n(G) =
χG(n)
nv(G)
.
8 B. Bychkov, A. Kazakov and D. Talalaev
Analogously, we express the partition function of the second model in terms of the flow poly-
nomial
CG(n) = (−1)e(G)+v(G)+k(G)TG(0, 1− n) =
(−1)e(G)+v(G)+k(G)−r(G)Z2
n(G)
nk(G)nv(G)−k(G)
= (−1)e(G)Z̃2
n(G). (2.2)
So we have
Z̃2
n(G) = (−1)e(G)CG(n). (2.3)
Then by Lemma 2.13 after the substitutions (2.2) and (2.3) we obtain
χG(n)
nv(G)
=
(n− 1)e(G)
ne(G)
∑
A′⊆G
CA′(n)
(1− n)e(A′)
,
where the summation goes through all edge induced subgraphs A′.
We finish the proof by noticing that the edge induced subgraph differs from the spanning
subgraph by the set of isolated vertices. Therefore we can complete each edge induced subgraph
to its corresponding spanning subgraph and then replace the summation over all edge induced
subgraph by the summation over all spanning subgraph, because the value of the each flow
polynomial CA′ remains the same and finally we obtain
χG(n) =
(n− 1)e(G)
ne(G)−v(G)
∑
A⊆G
CA(n)
(1− n)e(A)
. �
Note that we could produce series of statements that look like Theorem 2.18:
Theorem 2.19. Let us consider a graph G, then we can obtain the following formulas
nk(G)β
c(G)
1 (α1 − β1)r(G)TG
(
α1 + (n− 1)β1
α1 − β1
,
α1
β1
)
= qe(G)
∑
A⊆G
(
p
q
)e(A)
χA(n), (2.4)
where p = −α1 + β1, q = α1, and the summation (here and below) goes through all spanning
subgraphs A,
CG(n) = (n− 1)e(G)
∑
A⊆G
ne(A)−v(G)
(1− n)e(A)
χA(n), (2.5)
nk(G)−v(G)β
c(G)
1 (α1 − β1)r(G)TG
(
α1 + (n− 1)β1
α1 − β1
,
α1
β1
)
= q
e(G)
1
∑
A
(
p1
q1
)e(A)
(−1)e(A)CA(n), (2.6)
where p1 = β1−α1
n , q1 = α1−(1−n)β1
n ,
(−1)e(G)CG(n)
= q
e(G)
2
∑
A
(
p2
q2
)e(A)
nk(A)−v(G)β
c(A)
1 (α1 − β1)r(A)TA
(
α1 + (n− 1)β1
α1 − β1
,
α1
β1
)
, (2.7)
where p2 = n
β1−α1
and q2 = α1−(1−n)β1
α1−β1 .
Functional Relations on Anisotropic Potts Models 9
Proof. Let us consider two n-Potts models:
� Models M1(G;α1, β1) and M2(G; 0, 1) for the proof of the formula (2.4).
� The specification of the first case: M1(G; 1− n, 1) and the same M2(G; 0, 1) for the proof
of the formula (2.5).
� Models M1(G; 1− n, 1) and M2(G;α1, β1) with the parameters α1, β1 for the proof of the
formula (2.6).
� And finally, models M1(G;α1, β1) and M2(G; 1− n, 1) for the proof of formula (2.7).
Now it is left to repeat step by step the proof of Theorem 2.18 for these two models. �
Remark 2.20. We notice that the formula (2.5) naturally can be considered as “inversion”
of Theorem 2.18.
2.3 Shifting the order in the Potts models
Biggs Lemma 2.13 allows us to relate the values of the partition functions of the n-Potts models
with fixed n, but different values of parameters α and β. The goal of the current subsection
is to present a method for connecting partition functions of the n-Potts models for different n.
We will call it shifting order formulas.
The first method is based on the multiplicativity property of the complete flow polynomial.
Definition 2.21. Let G be a graph with the edge set E(G) and the vertex set V (G), let us
chose a fixed edge orientation on G. Then, a function f : E → Zn is called an n-flow if the
following condition holds
∀v ∈ V (G) :
∑
e∈M+(v)
f(e) =
∑
e∈M−(v)
f(e),
here again M+(v) (respectively M−(v)) is the set of edges each of them is directed to (respec-
tively from) v.
Let us formulate a few well known results concerning a flow polynomial and a number of all
n-flows. The proofs could be found for example in [22].
Proposition 2.22. Denote the number of all n-flows on a graph G by FCG(n), then FCG(n)
is independent of the choice of an orientation and the following identity holds
FCG(n) =
∑
A⊆G
CA(n),
where the summation goes through all spanning subgraphs A of the graph G.
Proposition 2.23. The number of all n-flows on a graph is the following polynomial (called
complete flow polynomial)
FCG(n) = ne(G)−v(G)+k(G),
where e(G), v(G), k(G) are numbers of edges, vertices and connected components in the graph G
correspondingly.
Proposition 2.24. The flow polynomial CG(n) of a graph G could be expressed in terms of the
complete flow polynomials of its spanning subgraphs by the following identity:
CG(n) =
∑
A⊆G
(−1)e(G)−e(A)FCA(n).
10 B. Bychkov, A. Kazakov and D. Talalaev
The complete flow polynomial FCG(n) is a multiplicative invariant: FCG(n1n2) = FCG(n1)
× FCG(n2), therefore we are ready to formulate the following theorem:
Theorem 2.25. The partition function Zn1n2(G) of the n1n2-Potts model M(G;α1, β1) could
be expressed in terms of the partition functions Zn1(A) and Zn2(A) of the n1-Potts model
M1(A;α1, β1) and n2-Potts model M2(A;α1, β1) of all spanning subgraphs A of the graph G
correspondingly.
Proof. Indeed, by Theorem 2.8 and the formula (2.6) we have
Zn1n2(G) = γGTG
(
α1 + (n1n2 − 1)β1
α1 − β1
,
α1
β1
)
=
∑
A⊆G
λACA(n1n2).
From Proposition 2.24 we obtain∑
A⊆G
λACA(n1n2) =
∑
A⊆G
λA
( ∑
A′⊆A
(−1)e(A)−e(A
′)FCA′(n1n2)
)
=
∑
A⊆G
ωAFCA(n1n2) =
∑
A⊆G
ωAFCA(n1)FCA(n2),
notice that we used for the second resummations the following simple observation: if X is a span-
ning subgraph of Y , which is a spanning subgraph of graph Z, so X is a spanning subgraph
of a graph Z. We omit this remark below.
The Proposition 2.22 implies∑
A⊆G
ωAFCA(n1)FCA(n2) =
∑
A⊆G
ωA
( ∑
A′⊆A
CA′(n1)
)( ∑
A′′⊆A
CA′′(n2)
)
=
∑
A′⊆G
∑
A′′⊆G
µA′A′′CA′(n1)CA′′(n2).
Finally, with the help of formula (2.7) and Theorem 2.8 we obtain∑
A′⊆G
∑
A′′⊆G
µA′A′′
( ∑
B⊆A′
δBZn1(B)
)( ∑
C⊆A′′
δCZn2(C)
)
=
∑
A′⊆G
∑
A′′⊆G
ηA′A′′Zn1(A′)Zn2(A′′),
where ηA′A′′ are some constants, appeared after the resummations. �
Remark 2.26 (convolution formula [16]). It seems extremely interesting and fruitful to compare
Lemma 2.13 and Theorem 2.25 with the convolution formula
TG(x, y) =
∑
A⊆E(G)
TG|A(0, y)TG/A(x, 0),
here the summation is over all possible subsets of E(G), here G|A is a graph obtained by the
restriction of G on the edge subset A and G/A is a graph obtained from G by the contraction
of all edges from A (see [9] for more details).
Our second method is based on the Tutte identity for the chromatic polynomial:
Theorem 2.27 ([9]). Consider a graph G with the set of edge V (G) then the following formula
holds
χG(n1 + n2) =
∑
B⊆V (G)
χG|B(n1)χG|Bc(n2),
where G|B (G|Bc) is the restriction of G on the vertex subset B ∈ V (G) (Bc ∈ V (G), where
Bc = V (G) \B).
Functional Relations on Anisotropic Potts Models 11
Using this fact we can formulate the following theorem:
Theorem 2.28. The partition function Zn1+n2(G) of the n1 + n2-Potts model M(G;α1, β1)
could be expressed in terms of the partition functions Zn1(A) and Zn2(A) of the n1-Potts model
M1(A;α1, β1) and n2-Potts model M2(A;α1, β1) of all spanning subgraphs A of the graph G
correspondingly.
Proof. The proof is very similar to the proof of Theorem 2.25. Again, from Theorem 2.8 and
the formula (2.4) we have
Zn1+n2(G) = γGTG
(
α1 + (n1 + n2 − 1)β1
α1 − β1
,
α1
β1
)
=
∑
A⊆G
λAχA(n1 + n2).
From Theorem 2.27 we obtain∑
A⊆G
λAχA(n1 + n2) =
∑
A⊆G
λA
( ∑
B⊆V (A)
χA|B(n1)χA|Bc(n2)
)
=
∑
A⊆G
λA
( ∑
B⊆V (A)
( ∑
A1⊆A|B
ωA1Zn1(A1)
)( ∑
A2⊆G|Bc
ωA2Zn2(A2)
))
=
=
∑
A⊆G
∑
B⊆V (A)
∑
A1⊆A|B
∑
A2⊆A|Bc
µA1A2Zn1(A1)Zn2(A2).
Let us complete each subgraph A1 (each A2) to the corresponding spanning subgraph of G by
adding isolating vertices∑
A⊆G
∑
B⊆V (A)
∑
A1⊆A|B
∑
A2⊆A|Bc
µA1A2Zn1(A1)Zn2(A2) =
∑
A′⊆G
∑
A′′⊆G
ηA′A′′Zn1(A′)Zn2(A′′),
where ηA′A′′ are again some constants, appeared after the resummations. �
3 Star-triangle equation for Ising and Potts models
3.1 General properties
Let us rewrite the partition function of the anisotropic n-Potts models in the so called Fortuin–
Kasteleyn representation:
Proposition 3.1 (compare with the formula (2.7) from [22]). Consider the anisotropic n-Potts
model M(G; ie), then its partition function could be expressed as follows
Zn(G) =
∑
σ
∏
e∈E
(βe + (αe − βe)δ(σe)) =
∏
e∈E
βe
∑
σ
∏
e∈E
(1 + (te − 1)δ(σe)), (3.1)
where δ(σe) is a value of standard Kronecker delta function of the values of σ on the boundary
vertices of the edge e and te = αe
βe
is a reduced weight of the edge e.
Proof. Indeed, it is easy to see that if σ(v) = σ(w):
ie(δ(e)) = ie(σ(v)− σ(w)) = αe = βe + (αe − βe)δ(σe) = βe + (αe − βe)δ(σ(v), σ(w)),
and if σ(v) 6= σ(w):
ie(δ(e)) = ie(σ(v)− σ(w)) = βe = βe + (αe − βe)δ(σe) = βe + (αe − βe)δ(σ(v), σ(w)). �
12 B. Bychkov, A. Kazakov and D. Talalaev
Also, we introduce the boundary partition function of the n-Potts models:
Definition 3.2. Let G be a graph (with possible loops and multiple edges) with the set of
vertices V , the set of edges E and the boundary subset S ⊆ V of enumerated vertices: S =
{v1, v2, . . . , vk}. The boundary partition function on G is defined by the following expression
Zn;S(A)(G) =
∑
σA
∏
e∈E
(βe + (αe − βe)δ(σe)),
where A = {a1, a2, . . . , ak}, ∀i : ai ∈ Zn is the set of fixed values, and the summation is over
such states σA that σA(vi) = ai.
Remark 3.3. If n = 2, we will omit the index 2 and will write just ZS(A)(G).
The next Lemma connects boundary and ordinary partition functions:
Lemma 3.4. Consider two graphs G1 = (V1, E1) and G2 = (V2, E2) with the only common
vertices in the boundary subset S = {v1, v2, . . . , vn} in V1 and V2. We can glue these graphs
and obtain the third graph G = (V,E), where E = E1 t E2, V = V1 ∪S V2. Then, the following
identity holds
Zn(G) =
∑
A
Zn;S(A)(G1)Zn;S(A)(G2),
where the summation is over all possible sets A.
v1
v2
v3v4
v1 v2
v3v4
w1 w2
w3
w4
Figure 3. G is obtained by merging of S = {v1, v2, v3, v4}.
Proof. The formula is obtained directly from the Proposition 3.1 and Definitions 3.2. Indeed,
by the Definition 3.2 we can write down Zn(G) =
∑
A
Zn;S(A)(G), but also Zn;S(A)(G) =
Zn;S1(A)(G1)Zn;S2(A)(G2). �
Remark 3.5. The latter property of a partition function (Lemma 3.4) allow us to consider
n-Potts model partition function as a discrete version of the topological quantum field theory
in the Atiyah formalism [1], where
TQFT : Cob→ Vect
is a functor from the category of cobordisms to the category of vector spaces.
Functional Relations on Anisotropic Potts Models 13
3.2 The case n = 2
In this subsection we consider the case n = 2. Our first goal is to find such conditions that
the partition function (3.1) is invariant under the star-triangle transformation which changes
the subgraph Ω to the subgraph Ω′. We derive these conditions with the use of the boundary
partition functions: consider a graph G with the subgraph Ω, then using Lemma 3.4 for graphs Ω
and G− Ω we obtain the following identity
Z(G) =
∑
A
ZS(A)(Ω)ZS(A)(G− Ω),
where S = {v1, v2, v3} (Figure 4). After the star-triangle transformation, we obtain a graph G′
with the following partition function
Z(G′) =
∑
A
ZS(A)(Ω
′)ZS(A)(G
′ − Ω′).
v1
v2 v2v3
v1
v3
t3
t0
3
t0
2 t0
1
t1 t2
Figure 4. Star-triangle transformation.
Due to the fact that the star-triangle transformation does not change edges of the graph G−Ω,
we deduce that ∀A : ZS(A)(G − Ω) = ZS(A)(G
′ − Ω′). Therefore, the sufficient and necessary
conditions for the invariance of the partition function are the following
∀A : ZS(A)(Ω) = ZS(A)(Ω
′). (3.2)
We write them down in detail. Let us note that these conditions do not depend on the states
of the vertices (see Figure 5), but depend on the number and the positions of the vertices with
equal states. Therefore, we have the following possibilities:
� two states in the triangle are the same, then the central vertex either has the same state
or has the different state, then α1β2β3 + β1α2α3 7→ α′1β
′
2β
′
3 and two more maps after
permuting indexes,
� all states are the same, then α1α2α3 + β1β2β3 7→ α′1α
′
2α
′
3.
In this way we obtain the following equations
α1β2β3 + β1α2α3 = α′1β
′
2β
′
3,
α2β1β3 + β2α1α3 = α′2β
′
1β
′
3,
α3β1β2 + β3α1α2 = α′3β
′
1β
′
2,
α1α2α3 + β1β2β3 = α′1α
′
2α
′
3.
(3.3)
14 B. Bychkov, A. Kazakov and D. Talalaev
a2
a1 a2
a2
a1 a2
a2
a2 a1
a2
a2 a1
a1
a2 a2
a1
a2 a2
a1
a1 a1
a1
a1 a1
1 2
3 4
Figure 5. Different possibilities.
After the substitution
ti =
αi
βi
we rewrite (3.3) as
β1β2β3(t1 + t2t3) = β′1β
′
2β
′
3t
′
1,
β1β2β3(t2 + t1t3) = β′1β
′
2β
′
3t
′
2,
β1β2β3(t3 + t1t2) = β′1β
′
2β
′
3t
′
3,
β1β2β3(t1t2t3 + 1) = β′1β
′
2β
′
3t
′
1t
′
2t
′
3.
This set of equations defines a correspondence which preserves the Ising model partition
function if we mutate the graph G to G′. Let us denote the product β1β2β3 by β and the
product β′1β
′
2β
′
3 by β′. Then, we obtain the following map from the (t, β)-variables to the (t′, β′)
variables, we will call it F̃ ,
F̃ (t1, t2, t3, β) = (t′1, t
′
2, t
′
3, β
′) :
t′1 =
√
(t1 + t2t3)(t1t2t3 + 1)
(t2 + t1t3)(t3 + t1t2)
,
t′2 =
√
(t2 + t1t3)(t1t2t3 + 1)
(t1 + t2t3)(t3 + t1t2)
,
t′3 =
√
(t3 + t1t2)(t1t2t3 + 1)
(t1 + t2t3)(t2 + t1t3)
,
β′ = β
√
(t1 + t2t3)(t3 + t1t2)(t2 + t1t3)
(t1t2t3 + 1)
. (3.4)
Remark 3.6. Formally speaking to define a map on the space of edge weight adopted to the
star-triangle transformation we have to resolve the map F̃ somehow for the parameters βi.
For example one can take the following one
β′i = βi(β
′/β)1/3.
Actually, the choice of a resolution is not important in what follows.
Remark 3.7. We choose the positive branch of the root function for real positive values of vari-
ables ti for purposes emphasized further. This is relevant to the almost positive version of the
orthogonal grassmanian. See [11] for more details about the connection of the Ising model and
positive orthogonal grassmanian.
Functional Relations on Anisotropic Potts Models 15
3.3 The case n 6= 2
Let us demonstrate how the method, described above, works for the star-triangle transformation
in the case n ≥ 3. Using the same ideas as in the previous subsection we could obtain the
following conditions
β1β2β3(t1 + t2t3 + n− 2) = β′1β
′
2β
′
3t
′
1,
β1β2β3(t2 + t1t3 + n− 2) = β′1β
′
2β
′
3t
′
2,
β1β2β3(t3 + t1t2 + n− 2) = β′1β
′
2β
′
3t
′
3,
β1β2β3(t1t2t3 + n− 1) = β′1β
′
2β
′
3t
′
1t
′
2t
′
3,
β1β2β3(t1 + t2 + t3 + n− 3) = β′1β
′
2β
′
3.
(3.5)
Here the last equation follows from the extra case in which all states are different.
a1
a3 a2
a1
a3 a2
Figure 6. The extra case.
In general, the system (3.5) does not have a solution and the star-triangle transformation
is not possible. But, if ti satisfy the special condition, partition function of the n-Potts model
is still invariant under the star-triangle transformation.
Proposition 3.8. The system (3.5) together with equation
t1t2t3 = t1t2 + t2t3 + t3t1 + (n− 1)(t1 + t2 + t3) + n2 − 3n+ 1 (3.6)
has a solution in terms of prime variables.
Proof. Using the first three and the last equations of (3.5) we immediately obtain the expres-
sions for t′i and
β′1β
′
2β
′
3
β1β2β3
:
β′1β
′
2β
′
3
β1β2β3
= t1 + t2 + t3 + n− 3,
t′1 =
t1 + t2t3 + n− 2
t1 + t2 + t3 + n− 3
,
t′2 =
t2 + t1t3 + n− 2
t1 + t2 + t3 + n− 3
,
t′3 =
t3 + t1t2 + n− 2
t1 + t2 + t3 + n− 3
.
Substitute these expressions into the fourth equation of (3.5) and obtain the equation
t1t2t3 + n− 1 =
(t1 + t2t3 + n− 2)(t2 + t1t3 + n− 2)(t3 + t1t2 + n− 2)
(t1 + t2 + t3 + n− 3)2
. (3.7)
By the straightforward computation, we retrieve that the identity (3.6) is the consequence
from the equation (3.7). �
Corollary 3.9. Partition function of the n-Potts model (n ≥ 3) is invariant under the star-
triangle transformation if and only if the system (3.5) with the equation (3.6) hold.
16 B. Bychkov, A. Kazakov and D. Talalaev
Below we present two nontrivial specialization of the partition function of n-Potts model
which are agreed with the system (3.5), (3.6).
Example 3.10. Consider a graph G and equip each e ∈ E(G) with sign + or −. Let us consider
the n-Potts model Mk(G,αe, βe) with following parameters:
� for all e ∈ E equipped with + the parameters αe, βe equal αe = A+ = −t−
3
4 , βe = B+ = t
1
4 ,
� for all e ∈ E equipped with− the parameters αe, βe equal αe = A− = −t
3
4 , βe = B− = t−
1
4 ,
� and n = t+ 1
t + 2 (we suppose that parameter t is chosen such that n ∈ N).
Let the graph G has a triangle subgraph, the edges of which have signs +, −, −. The reduced
weights of edges are t1 = A+
B+
= −1
t , t2 = A−
B−
= −t, t3 = A−
B−
= −t. It is easy to see that these ti
satisfy the equation (3.6).
We notice that the signed graph G could be considered as the signed Tait graph for a dia-
gram D(K) of a knot K ([24], the chapter “Knot invariants from edge-interaction models”).
Moreover, the value of the Jones polynomial of the knot K at the point n is closely related
with the partition function of the n-Potts model Mk(G,αe, βe) (see [24, equation (7.17)]). Thus,
the identification of the third Reidemeister move of the diagram D(K) with the star-triangle
transformation of the signed graph G is agreed with the star-triangle transformation defined
by the system (3.5), (3.6) for the n-Potts model Mk(G,αe, βe).
Our second example is about the models of bond percolation. Firstly, we briefly give their
definitions:
Definition 3.11 (bond percolation [13]). Consider a graph G. An edge e ∈ E(G) is considered
to be open with probability pe or closed with probability 1 − pe. We suppose that all edges
might be closed or open independently. One is interested in probabilistic properties of cluster
formation (i.e. maximal connected sets of closed edges of the graph G).
Example 3.12. The bond percolation models could be considered as a limit n → 1 of the n-
Potts models at the level of the boundary partition functions [7]. This identification corresponds
to the specialization of the system (3.5), (3.6) by n→ 1.
Substitute ti = 1
pi
, t′i = 1
p′i
and n = 1 in (3.5) and (3.6), then
(p1 + p2p3 − p1p2p3)α1α2α3 = p′2p
′
3α
′
1α
′
2α
′
3,
(p2 + p1p3 − p1p2p3)α1α2α3 = p′1p
′
3α
′
1α
′
2α
′
3,
(p3 + p1p2 − p1p2p3)α1α2α3 = p′1p
′
2α
′
1α
′
2α
′
3,
α1α2α3 = α′1α
′
2α
′
3,
1
p1p2
+
1
p2p3
+
1
p1p3
− 1 =
1
p1p2p3
.
After simplifications we obtain the condition for the star-triangle transformation of the bond
percolation models (for instance, see [13])
p1 + p2p3 − p1p2p3 = p′2p
′
3,
p2 + p1p3 − p1p2p3 = p′1p
′
3,
p3 + p1p2 − p1p2p3 = p′1p
′
2,
p1 + p2 + p3 − 1 = p1p2p3.
Functional Relations on Anisotropic Potts Models 17
4 Tetrahedron equation
The tetrahedron equation firstly was considered by A. Zamolodchikov [25] who has constructed
its solution in S-form. We consider the following form of the equation
T123T145T246T356 = T356T246T145T123, (4.1)
where Tijk is an operator acting nontrivially in the tensor product of three vector spaces Vi,
Vj , Vk, indexed by i, j and k. Tetrahedron equation is the higher order analog of the Yang–
Baxter equation. Both equations are examples of n-simplex equations [18] and play an important
role in hypercube combinatorics and higher Bruhat orders. For the complete introduction to
the topic see for example [21]. In this section we present two proofs of the main theorem of the
paper:
Theorem 4.1. The change of variables (3.4) defines the solution of the tetrahedron equa-
tion (4.1).
These two proofs have a lot of common points and ideas, but have the crucial differences
in the last stages. It is interesting to compare proofs for the purpose of combining arguments
of boundary partition functions and the technique of correlation functions in the Ising–Potts
models.
At first in the next subsection we prove that the change of variables (3.4) corresponds to the
variables transform in the trigonometric solution of a local Yang–Baxter equation.
4.1 Local Yang–Baxter equation
Let us recall that the following change of variables (t1, t2, t3) 7→ (t′1, t
′
2, t
′
3) provides an invariance
of the Ising model (3.1) under the star-triangle transformation (3.4):
t′1 =
√
(t1 + t2t3)(t1t2t3 + 1)
(t2 + t1t3)(t3 + t1t2)
,
t′2 =
√
(t2 + t1t3)(t1t2t3 + 1)
(t1 + t2t3)(t3 + t1t2)
,
t′3 =
√
(t3 + t1t2)(t1t2t3 + 1)
(t1 + t2t3)(t2 + t1t3)
m
t′1t
′
2 =
t1t2t3 + 1
t3 + t1t2
,
t′2t
′
3 =
t1t2t3 + 1
t1 + t2t3
,
t′1t
′
3 =
t1t2t3 + 1
t2 + t1t3
.
(4.2)
Following [17], we construct orthogonal hyperbolic 3 × 3 matrices Rij which solve the local
Yang–Baxter equation
R12(t3)R13(S(t2))R23(t1) = R23(S(t′1))R13(t
′
2)R12(S(t′3)), (4.3)
where S(t) is the following involution
S(t) =
t− 1
t+ 1
. (4.4)
18 B. Bychkov, A. Kazakov and D. Talalaev
On the left hand side of (4.3) we have
R12(t3) =
i sinh(log(t3)) cosh(log(t3)) 0
cosh(log(t3)) −i sinh(log(t3)) 0
0 0 1
, (4.5)
R13(S(t2)) =
i sinh(log(S(t2))) 0 cosh(log(S(t2)))
0 1 0
cosh(log(S(t2))) 0 −i sinh(log(S(t2)))
, (4.6)
R23(t1) =
1 0 0
0 i sinh(log(t1)) cosh(log(t1))
0 cosh(log(t1)) −i sinh(log(t1))
. (4.7)
Theorem 4.2. Matrices (4.5), (4.6), (4.7) together with the rules (3.4), (4.4) give a solution
of (4.3).
Proof. It can be proved by a straightforward computation. For example let us write down the
result of the product on the left hand side
t2
(
t23 − 1
)
t3(t22 − 1)
i
(
t21t
2
2t
2
3 − t21 − t22 + t23
)
2t1t3
(
t22 − 1
) t21t
2
2t
2
3 − t21 + t22 − t23
2t1t3
(
t22 − 1
)
−
it2
(
t23 + 1
)
t3(t22 − 1)
t21t
2
2t
2
3 + t21 + t22 + t23
2t1t3
(
t22 − 1
) −
i
(
t21t
2
2t
2
3 + t21 − t22 − t23
)
2t1t3
(
t22 − 1
)
t22 + 1
t22 − 1
it2
(
t21 + 1
)
t1
(
t22 − 1
) t2
(
t21 − 1
)
t1
(
t22 − 1
)
. (4.8)
At the first glance the product on the right hand side looks much more cumbersome, but occa-
sionally all terms are simplified and the matrix on the right hand side coincides with (4.8). �
4.2 Tetrahedron equation, first proof
Let us encode the tetrahedron equation by the Figure 7.
a
b
c
da
b
c
1
2
3
4
5
6
d
Figure 7. Encoding the tetrahedron equation by the standard graph.
The standard graph encodes R-matrices in the following way: in each inner vertex numbered
by k, which is the intersection of strands i and j we put the matrix Rij(tk) which is the 2 × 2
matrix in the 4-dimensional space with basis vectors indexed by a, b, c, d. For instance,
Rac(t5) =
i sinh(log(t5)) 0 cosh(log(t5)) 0
0 1 0 0
cosh(log(t5)) 0 −i sinh(log(t5)) 0
0 0 0 1
.
Functional Relations on Anisotropic Potts Models 19
Let us orient each strand from the left to the right and multiply R-matrices in order of the
orientation, for instance for the Figure 7 we have the following product of R-matrices
Rcd(t1)Rbd(S(t2))Rbc(t3)Rad(t4)Rac(S(t5))Rab(t6). (4.9)
We note that the orientation defines the product (4.9) uniquely.
Then let us apply four local Yang–Baxter equations consequently to the inner triangles with
vertices numbered (1, 2, 3), (1, 4, 5), (2, 4, 6) and (3, 5, 6) as on the Figure 8. As a result, we have
one and the same standard graph as on the Figure 7 rotated by π.
a
b
c
da
b
c
1
2
3
4
5
6
d
a
b
c
d a
b
c
1
2
3
4
5
6
d
a
b
c
da
b
c
1
23
4
5
6
d
c
d
a
a
b
b
c
1
2
3
4
5
6
d
a
b
c
d
a
b
c
1
2
3
4
5
6
d
T123
T145
T246T356
Figure 8. Local Yang–Baxter equations applied to the standard graph.
At the same time we could apply local Yang–Baxter equations in the opposite direction:
firstly to the triangle (3, 5, 6), then (2, 4, 6), (1, 4, 5) and (1, 2, 3). Eventually in this case we will
have again the same standard graph.
As the reader may have already guessed, every local Yang–Baxter equation applied to the
triangle A, B, C defines the factor TABC in the tetrahedron equation (4.1). For example we
obtain
T1,2,3 : (t1, S(t2), t3, t4, t5, t6) 7→ (S(t′1), t
′
2, S(t′3), t4, t5, t6).
By Theorem 4.2 the product (4.9) preserves by each local Yang–Baxter equation encoded on the
Figure 8. As a result of two sequences of four Local Yang–Baxter equations we obtain an equality
of two products of six 4× 4 R-matrices
Rcd(u1)Rbd(u2)Rbc(u3)Rad(u4)Rac(u5)Rab(u6)
= Rcd(v1)Rbd(v2)Rbc(v3)Rad(v4)Rac(v5)Rab(v6), (4.10)
where the parameters ui, vj , i, j = 1, . . . , 6 depend on the initial variables ti, on the mapping (3.4)
and on the involution (4.4).
20 B. Bychkov, A. Kazakov and D. Talalaev
Let us consider this equation element-wise, and note that we could uniquely express parame-
ters in the right hand side in terms of the parameters on the left hand side
U1,4 = b(t4), U2,4 = −a(t4)b(t2), U1,3 = a(t4)b(t5),
U1,2 = a(t4)a(t5)b(t6), U3,4 = a(t2)a(t4)b(t1), U2,3 = b(t2)b(t4)b(t5)− a(t2)a(t5)b(t3).
Here U is a matrix on the left hand side and a, b are some invertible functions, come from (4.5),
(4.6), (4.7). So we could uniquely determine t4 from the first equation, t2 and t5 from the second
and the third, then t1 and t6, and finally t3 from the element U2,3.
Let us note that this algebraic proof could be formulated in terms of the paths on the standard
graph (Figure 7) with orientation. So the equation (4.10) provides coincidence of the parameters
in the vertices given by the two sides of the tetrahedron equation. This finishes the proof.
4.3 Tetrahedron equation, second proof
4.3.1 Involution lemma
Let us consider the map F (t1, t2, t3) = (t′1, t
′
2, t
′
3), where prime variables defined by (3.4). First
of all, we formulate one technical lemma:
Lemma 4.3. The following identity holds for all t1, t2, t3:
S × S × S ◦ F ◦ S × S × S = F−1,
where S(t) = t−1
t+1 .
We present the proof in Appendix A.
4.3.2 Towards the tetrahedron equation
Let us consider any graph G with a subgraph Γ1 which coincides with the leftmost graph on the
Figure 9. We can transform the graph G to the graph G′ with the subgraph Γ2 which coincides
with the rightmost graph on the Figure 9. We could make this mutation by two different chains
of star-triangle transformations: F−1356F246F
−1
145F123 and F−1123F145F
−1
246F356. Both are figured out
on the Figure 8. This observation turns us to the following hypothesis
F−1356F246F
−1
145F123 = F−1123F145F
−1
246F356. (4.11)
1
3 3
3
45
1
2
6
2 5
5 2
6 1
1 6
4 4
2
65
4 3
56
21
3 4
1
3 3
3
4
6
2
1
5
1 6
6 1
5 2
2 5
4 4
2
65
4 3
56
21
3 4
Figure 9. The graphical representation of the left and right parts of (4.12).
Functional Relations on Anisotropic Potts Models 21
This equality is equivalent to the Zamolodchikov equation
Φ356Φ246Φ145Φ123 = Φ123Φ145Φ246Φ356, (4.12)
where Φijk = SiSkFijkSj . Indeed, using Lemma 4.3 and the simple observation that SlFijk =
FijkSl, l 6= {i, j, k} we can write down
F−1356F246F
−1
145F123 = S3S5S6F356S3S5S6F246S1S4S5F145S1S4S5F123
= S2S5(S3S6F356S5)(S2S6F246S4)(S1S5F145S4)(S1S3F123S2)S2S5,
and
F−1123F145F
−1
246F356 = S1S2S3F123S1S2S3F145S2S4S6F246S2S4S6F356
= S2S5(S1S3F123S2)(S1S5F145S4)(S2S6F246S4)(S1S3F356S2)S2S5.
Conjugating both sides of (4.11) by S2S5 we obtain the Zamolodchikov tetrahedron equation.
4.3.3 Solution for the tetrahedron equation
Proposition 4.4. The functions
�
∂ ln(Z(G))
∂te
, where e is any edge belonging to G− Ω, and
�
ZS(A)(G)
Z(G)
, where ZS(A)(G) is the boundary partition function and S is any vertex subset
of G− Ω or G− Ω′ (see the Figure 4),
are invariant under the star-triangle transformation. Moreover, these functions do not depend
on variables βe.
Remark 4.5. The function
ZS(A)(G)
Z(G)
can be interpreted as a probability of the fixed values A of spins in S, related to the boundary
partition function ZS(A)(G).
Proof. The crucial point in the demonstration of the first part of the statement is the fact that
the derivative ∂ ln(Z(G))
∂tei
does not depend on parameters β. Indeed, this follows from the explicit
form of the partition function
Z(G) =
∏
e∈E
βe
∑
σ
∏
e∈E
(1 + (te − 1)δ(σe)).
The proof of the second part of the statement is straightforward. It follows from the Definition 3.2
and the condition (3.2). �
We will prove the Zamolodchikov equation in its equivalent form
F−1356F246F
−1
145F123 = F−1123F145F
−1
246F356. (4.13)
Let us notice that due to the local nature of the star-triangle transformation and the convolution
property of the boundary partition function we have a choice to take some suitable graph to
22 B. Bychkov, A. Kazakov and D. Talalaev
v2 v1
2
v3 v4
v6
v5
v2 v1
v3 v4
v6
v5
1
Figure 10. The graphs Γ1 and Γ2.
prove equation (4.13). So let us take the graph Γ1 from the Figure 10 with the following choice
of boundary set S0:
S0 := {v1, v2, v3, v4}.
We will prove that the values of the second-type invariant functions which are preserved by
both sides of the equation (4.13) allows us to uniquely reconstruct weights of all edges. Explain
this idea in detail, let us consider the left hand side of the equation (4.13) and the map F123,
then for any A = {a1, . . . , a4} the following identity holds
ZS0(A)(Γ1)
Z(Γ1)
=
ZS1(A1)(Γ1)
Z(Γ1)
+
ZS1(A2)(Γ1)
Z(Γ1)
,
here S1 = {v1, v2, v3, v4, v5} (see Figure 11), A1 = {a1, . . . , a4, 0}, A2 = {a1, . . . , a4, 1}. The
Proposition 4.4 provides
ZS1(A1)(Γ1)
Z(Γ1)
=
ZS1(A1)(Γ
′)
Z(Γ′)
,
ZS1(A2)(Γ1)
Z(Γ1)
=
ZS1(A2)(Γ
′)
Z(Γ′)
,
where Γ′ is obtained from Γ1 by the star-triangle transformation (see Figure 11). And therefore
we deduce that
ZS0(A)(Γ1)
Z(Γ1)
=
ZS0(A)(Γ
′)
Z(Γ′)
.
Repeating these arguments for the remaining maps Fijk from the left hand side of (4.13) we
obtain
ZS0(A)(Γ1)
Z(Γ1)
=
ZS0(A)(Γ2)
Z(Γ2)
. (4.14)
In the same fashion, if we consider the right hand side of the equation (4.13), we similarly obtain
that
ZS0(A)(Γ1)
Z(Γ1)
=
ZS0(A)(Γ2)
Z(Γ2)
.
Hence, in order to prove the equation (4.13) it is sufficient to prove that we can reconstruct
the parameters ti, i = 1, . . . , 6 from the values ZS0(A)(Γ2)/Z(Γ2) for different values of A in
a unique way.
We understand the identity (4.14) as a system of 24 linear equations with unknowns
ZS0(A)(Γ2) of the following type
ZS(A)(Γ2) :
ZS(A)(Γ2)
Z(Γ2)
= α(A), ∀A = (a1, . . . , a4)
Functional Relations on Anisotropic Potts Models 23
v1 v2
v3v4
a
1
a
2
a
2
a
1
a3a4
a4
a4
a1 a2 a2a1
a4a4a4 a3
a3
a1 a2
a3
v5
1
1
1 0
0
1
1
1
1
0
0
0
0 1 1 0 1 0 0
0
1
1
0
1
0
1 0
0
F123 F145
–1
F356
–1
F246
Figure 11. The left hand side of (4.13).
v
2
v
1
v4v3
v5
t
1
t2
t5 t3
t4
t
6
v6
2
Figure 12. The graph Γ2.
which is equivalent to∑
A′
ZS(A′)(Γ2) = ZS(A)(Γ2)/α(A).
The rank of the system is equal to 15. Indeed, the rank is ≥ 15 and we know that there is
a nontrivial solution coming from the boundary partition functions for the graph Γ2.
Hence any solution has the form
ZS0(A)(Γ2) = C · α0(a1, a2, a3, a4), (4.15)
where C is some constant and ai are the states. Now we will prove that the parameters t1, . . . , t6
are reconstructed uniquely from the equation (4.15).
Let us introduce some auxiliary variables and rewrite the partition function in the following
way: we have 16 states of boundary vertices S0 = {v1, v2, v3, v4}. Each expression ZS0(A)(Γ2)
is a sum of four terms corresponding to the states of internal vertices v5 and v6. We consider
24 B. Bychkov, A. Kazakov and D. Talalaev
in details the case S0 = {0, 0, 0, 0}. Let us denote the weights of the states of the square
{v1, v6, v5, v4} by v, z, y and x (Figure 13). Then, we obtain the following equations
v = t3t5B, v1 = t5t6B,
z = t3t4t5t6B, z1 = t4t5B,
y = t6t3B, y1 = B,
x = t3t4B, x1 = t4t6B, (4.16)
where B = β3β4β5β6 and
v + t1t2z + t2y + t1x =
C
B1
α0(0, 0, 0, 0), B1 = β1β2.
0
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
1
1
0
1
0
1
0
1
1 1 1 1
v5
v4
v6
v1
vzyx
vzyx
Figure 13. The auxiliary variables.
Similarly we obtain seven more equations (we omit eighth equation with v1 = 1 due to the
symmetry of the model with respect to the total involution of spins)
v1 + t1t2z1 + t2y1 + t1x1 =
C
B1
b1 =
C
B1
α′(0, 0, 0, 1),
t1v1 + t2z1 + y1t1t2 + x1 =
C
B1
b2 =
C
B1
α′(0, 1, 0, 1),
t1t2v1 + z1 + t1y1 + t2x1 =
C
B1
b3 =
C
B1
α′(0, 1, 1, 1),
t2v1 + t1z1 + y1 + t1t2x1 =
C
B1
b4 =
C
B1
α′(0, 0, 1, 1),
v + t1t2z + t2y + t1x =
C
B1
a1 =
C
B1
α′(0, 0, 0, 0),
t1v + t2z + yt1t2 + x =
C
B1
a2 =
C
B1
α′(0, 1, 0, 0),
t1t2v + z + t1y + t2x =
C
B1
a3 =
C
B1
α′(0, 1, 1, 0),
t2v + t1z + y + t1t2x =
C
B1
a4 =
C
B1
α′(0, 0, 1, 0).
By straightforward calculation we retrieve
y1 =
C
B1
−t2b1 + t1t2b2 + b4 − t1b3
−t22 + 1 + t21t
2
2 − t21
, z1 =
C
B1
t1t2b1 − t2b2 + b3 − t1b4
−t22 + 1 + t21t
2
2 − t21
,
x1 =
C
B1
t2t1b4 − t1b1 − t2b3 + b2
−t22 + 1 + t21t
2
2 − t21
, v1 =
C
B1
b1 + t1t2b3 − t2b4 − t1b2
−t22 + 1 + t21t
2
2 − t21
, (4.17)
Functional Relations on Anisotropic Potts Models 25
y =
C
B1
−t2a1 + t1t2a2 + a4 − t1a3
−t22 + 1 + t21t
2
2 − t21
, z =
C
B1
t1t2a1 − t2a2 + a3 − t1a4
−t22 + 1 + t21t
2
2 − t21
,
x =
C
B1
t2t1a4 − t1a1 − t2a3 + a2
−t22 + 1 + t21t
2
2 − t21
, v =
C
B1
a1 + t1t2a3 − t2a4 − t1a2
−t22 + 1 + t21t
2
2 − t21
.
Using the auxiliary variables it is easy to see that
z1
x1
=
v
y
=
−t2a4 + t1t2a3 − t1a2 + a1
t2t1a2 − t2a1 − t1a3 + a4
=
t1t2b1 − t2b2 + b3 − t1b4
t2t1b4 − t1b1 − t2b3 + b2
.
In the same fashion we obtain
v1
x1
=
v
x
.
Then we can straightforwardly deduce expressions for the variables t1 and t2 (equations (B.1),
(B.2) in Appendix B), then obtain expressions for the auxiliary variables from equations (4.17)
and finally obtain variables t3, t4, t5, t6 from the equations (4.16):
t3 =
√
vyx
zy21
,
t5 =
v
t3y1
,
t6 =
y
t3y1
,
t4 =
x
t3y1
.
This completes the proof of the Zamolodchikov equation due to the fact that there is a uni-
que way to choose positive weights for the edges of the model to provide the expected values
of boundary partitions function for the graph Γ2.
5 Star-triangle transformation, Biggs formula and conclusion
The main results of the paper represent the functional relations on the space of multivariate
Tutte polynomials. This problem is a step of the program of investigation of the framed graph
structures and the related statistical models. We examined in details the Biggs formula and
applied it to the multivariate case. We also provided a new proof of the theorem of Matiyasevich
as a partial case of such formula. The second principal result is the reveal of the tetrahedral
symmetry of the multivariate Tutte polynomial at the point n = 2. Therefore, we have a connec-
tion between the multivariate Tutte polynomial, functions on Lustig cluster manifolds [4] and
its electrical analogues [12, 19]. We would like to interpret this property as the critical point
of the model described by the multivariate Tutte polynomial, and the tetrahedral symmetry as
a longstanding analog of the conformal symmetry of the Ising model at the critical point [8].
Both correspondences are related by the following observation. Let G and G′ be two graphs
related by the star-triangle transformation. Let us consider the case when the partition function
is invariant with respect to this transformation (n = 2 or n > 2 and the system (3.5), (3.6)
holds)
Zn(G′) = Zn(G).
On the other hand the star-triangle transformation provides a groupoid symmetry on a wide
class of objects, in our case on the space of Ising models. The Biggs formula allows us to
extend this action to the points of valency 1 and 2. And we can obtain the 14-term relation
(Theorem 5.1) by comparing the right-hand sides of Biggs formulas for G and G′.
26 B. Bychkov, A. Kazakov and D. Talalaev
Let us explain this idea in details, consider two pairs of n-Potts models: M1(G, i
1
e) and
M1(G
′, i1e), M2(G, i
2
e) and M2(G
′, i2e) (for simplicity we denote these models M1(G), M2(G),
M1(G
′), M2(G
′) correspondingly). After multiplying both parts of the formula (2.1) for M1(G)
and M2(G) by nv(G) (by nv(G
′) for M1(G
′) and M2(G
′)) we obtain
Z1
n(G) =
∏
e∈G
qe
∑
A⊆G
∏
e∈A
pe
qe
Z2
n(A) = Z1
n(G′) =
∏
e∈G′
q′e
∑
A′⊆G′
∏
e∈A′
p′e
q′e
Z2
n(A′), (5.1)
here in both cases we take the sum over the set of all spanning subgraphs.
Let us rewrite the first part of the formula (5.1) by separating two kinds of terms
Z1
n(G) =
∏
e∈G
qe
∑
A1⊆G
∏
e∈A1
pe
qe
Z2
n(A1) +
∏
e∈G
qe
∑
A2⊆G
∏
e∈A2
pe
qe
Z2
n(A2), (5.2)
where each subgraph A1 contains the full triangle and each A2 contains only a part of the
triangle.
After the star-triangle transformation of the M1(G) and M2(G) we obtain the following
formula for the models M1(G
′) and M2(G
′):
Z1
n(G′) =
∏
e∈G′
q′e
∑
A′1⊆G′
∏
e∈A′1
p′e
q′e
Z2
n(A′1) +
∏
e∈G′
q′e
∑
A′2⊆G′
∏
e∈A′2
p′e
q′e
Z2
n(A′2), (5.3)
where each subgraph A′1 contains the full star and each A′2 contains only a part of the star.
Then, we compare the terms of these formulas:
� We notice that due to the star-triangle transformation Zin(G) = Zin(G′) and Zin(A1) =
Zin(A′1) (here and below A1 is different from A′1 only by the star-triangle transformation).
� Also it is easy to see that
∏
e∈G qe
1∏
e∈A1
qe
=
∏
e∈G′ q
′
e
1∏
e∈A′1
q′e
.
� If the model M1(G) is chosen such that p1p2p3 = p′1p
′
2p
′
3, we conclude that
∏
e∈A1
pe =∏
e∈A′1
p′e.
Now, we are ready to formulate the following theorem:
Theorem 5.1. Consider two n-Potts models M2(G) and M2(G
′), which are different from each
other by the star-triangle transformation. Then, the following formula holds
q1q2q3
(
Z2
n(G0) +
p1
q1
Z2
n(G1) +
p2
q2
Z2
n(G2) +
p3
q3
Z2
n(G3) +
p1p2
q1q2
Z2
n(G12) +
p1p3
q1q3
Z2
n(G13)
+
p2p3
q2q3
Z2
n(G23)
)
= q′1q
′
2q
′
3
(
Z2
n(G′0) +
p′1
q′1
Z2
n(G′1) +
p′2
q′2
Z2
n(G2) +
p′3
q′3
Z2
n(G′3)
+
p′1p
′
2
q′1q
′
2
Z2
n(G′12) +
p′1p
′
3
q′1q
′
3
Z2
n(G′13) +
p′2p
′
3
q′2q
′
3
Z2
n(G′23)
)
, (5.4)
where
pi =
α1
i − β1i
α2
i − β2i
, qi =
α2
i β
1
i − α1
i β
2
i
α2
i − β2i
, p′i =
α′1i − β′1i
α′2i − β′2i
, q′i =
α′2i β
′1
i − α′1i β′2i
α′2i − β′2i
,
variables αki , βki and α′ki , β′ki are related by the star-triangle transformation with the condition
p1p2p3 = p′1p
′
2p
′
3 and graphs Gi and Gij, i, j = 0, 1, 2, 3 are depicted on the Figure 14.
Functional Relations on Anisotropic Potts Models 27
G
13
G
23
G
12
G
3
G
2
G
2
G
1
G
3
G
12
G
13
G
23
G
0
G
1
G
0
Figure 14. The 14-term relation.
Proof. We will prove this theorem using induction on ex(G) := e(G)− 3.
We show that the base of the induction k = 0 is trivial. Hence, let us consider the n-Potts
modelsM2(G) andM2(G
′) and the special modelsM1(G) andM1(G
′) such that p1p2p3 = p′1p
′
2p
′
3.
Then, we write the formulas (5.2) and (5.3), after the comparison for each terms using the
reasoning above we immediately obtain the result in the case ex(G) = 0.
Then, make the step of induction. Again, let us write down the formulas (5.2) and (5.3):
Z1
n(G) =
∏
e∈G
qe
∑
A1⊆G
∏
e∈A1
pe
qe
Z2
n(A1) +
∏
e∈G
qe
∑
A2⊆G
∏
e∈A2
pe
qe
Z2
n(A2) +
∏
e∈G qe
q1q2q3
S1,
where each A1 contains the full triangle, each A2 contains only a part of the triangle and such
that ex(A2) 6= ex(G), and by S1 we denoted the left hand side of (5.4),
Z1
n(G′) =
∏
e∈G′
q′e
∑
A′1⊆G′
∏
e∈A′1
p′e
q′e
Z2
n(A′1) +
∏
e∈G′
q′e
∑
A′2⊆G′
∏
e∈A′2
p′e
q′e
Z2
n(A′2) +
∏
e∈G′ q
′
e
q′1q
′
2q
′
3
S2,
where each A′1 contains the full star, each A′2 contains only a part of the star and such that
ex(A′2) 6= ex(G′), and by S2 we denoted the right hand side of (5.4).
Then the induction assumption ends the proof. �
We consider these results in the context of numerous generalizations, both for other models
of statistical physics, and in a purely mathematical direction. In particular, we are interested
in applying this technique to the Potts model in the presence of an external magnetic field [10],
including an inhomogeneous one. In addition, we are going to develop these methods in a more
general algebraic sense, in particular in a non-commutative situation. Partial results of this
activity have already been obtained in [6].
A The proof of Lemma 4.3
Proof. We start by reformulating this statement in terms of equivalent rational identities. Let
us introduce the x-variables by the following formula
F ◦ S3(t1, t2, t3) = (x1, x2, x3),
S3(x1, x2, x3) = (t′1, t
′
2, t
′
3). (A.1)
28 B. Bychkov, A. Kazakov and D. Talalaev
Here S3(t1, t2, t3) = (S × S × S)(t1, t2, t3) = (S(t1), S(t2), S(t3)). Then the statement of the
lemma is equivalent to
(t′1, t
′
2, t
′
3) = S3(x1, x2, x3) = F−1(t1, t2, t3).
This identity is equivalent to three algebraic relations (we will write down only one of them,
because the others differ just by replacing the indices)
t1t2 =
t′1t
′
2t
′
3 + 1
t′3 + t′1t
′
2
=
(
(x1 − 1)(x2 − 1)(x3 − 1)
(x1 + 1)(x2 + 1)(x3 + 1)
+ 1
)/(
x3 − 1
x3 + 1
+
(x1 − 1)(x2 − 1)
(x1 + 1)(x2 + 1)
)
=
x1 + x2 + x3 + x1x2x3
x3 − x2 − x1 + x1x2x3
=
(x1 + x2 + x3 + x1x2x3)x1x2x3
(x3 − x2 − x1 + x1x2x3)x1x2x3
. (A.2)
Now let us introduce some additional variables
t12 = x1x2, t23 = x2x3, t13 = x1x3, a1 = x21, a2 = x22, a3 = x23.
We could rewrite (A.2) in the following way
t1t2 =
a1t23 + a2t13 + a3t12 + t12t23t13
−a2t13 − a1t23 + a3t12 + t12t23t13
.
The equations (4.2) and (A.1) with the identification yi := S(ti) provide the following system
t12 =
y1y2y3 + 1
y3 + y1y2
=
t3 + t2 + t1 + t1t2t3
t3 − t2 − t1 + t1t2t3
,
t13 =
y1y2y3 + 1
y2 + y3y1
=
t3 + t2 + t1 + t1t2t3
t2 − t1 − t3 + t1t2t3
,
t23 =
y1y2y3 + 1
y1 + y3y2
=
t3 + t2 + t1 + t1t2t3
t1 − t2 − t3 + t1t2t3
,
a1 =
(y1y2y3 + 1)(y1 + y2y3)
(y2 + y1y3)(y3 + y1y2)
=
(t1 − t2 − t3 + t1t2t3)(t3 + t2 + t1 + t1t2t3)
(t3 − t2 − t1 + t1t2t3)(t2 − t1 − t3 + t1t2t3)
,
a2 =
(y1y2y3 + 1)(y2 + y1y3)
(y1 + y2y3)(y3 + y1y2)
=
(t2 − t1 − t3 + t1t2t3)(t3 + t2 + t1 + t1t2t3)
(t3 − t2 − t1 + t1t2t3)(t1 − t2 − t3 + t1t2t3)
,
a3 =
(y1y2y3 + 1)(y3 + y2y1)
(y2 + y1y3)(y1 + y3y2)
=
(t3 − t2 − t1 + t1t2t3)(t3 + t2 + t1 + t1t2t3)
(t1 − t2 − t3 + t1t2t3)(t2 − t1 − t3 + t1t2t3)
.
Using these expressions we can compute
t12a3 + t13a2 + a1t23 + t12t23t13
=
4(t3 + t2 + t1 + t1t2t3)
2t1t2t3
(t3 − t2 − t1 + t1t2t3)(t2 − t1 − t3 + t1t2t3)(t1 − t2 − t3 + t1t2t3)
,
t1t2(−a2t13 − a1t23 + a3t12 + t12t23t13)
=
4(t3 + t2 + t1 + t1t2t3)
2t1t2t3
(t3 − t2 − t1 + t1t2t3)(t2 − t1 − t3 + t1t2t3)(t1 − t2 − t3 + t1t2t3)
.
In this way we observe that
t1t2(−a2t13 − a1t23 + a3t12 + t12t23t13) = a3t12 + a2t13 + a1t23 + t12t23t13.
This completes the proof. �
Functional Relations on Anisotropic Potts Models 29
B For the second proof of the tetrahedron equation
In this Appendix we present some technical part of the proof from Section 4.3. We present
closed formulas for t1 and t2 variables (see the discussion after (4.17))
t1 =
(
−b3a3 + a2b2 + a1b1 − b4a4 + (b23a
2
3 − 2b3a3a2b2 − 2b3a3a1b1 − 2b3a3b4a4
+ a22b
2
2 − 2a2b2a1b1 − 2a2b2b4a4 + a21b
2
1 − 2a1b1b4a4 + b24a
2
4 + 4b4a3a1b2
+ 4a2b1b3a4)
1/2
)
/(2(−b4a3 + a2b1)), (B.1)
t2 = (a2b4 − b2a4 − b3a1 + a3b1 + (a22b
2
4 − 2a4b4a2b2 − 2a2b4b3a1 − 2a2b1a3b4
+ b22a
2
4 − 2b3a4a1b2 −−2b2a4a3b1 + b23a
2
1 − 2a3b3a1b1 + a23b
2
1 + 4a3b4a1b2
+ 4a2b1b3a4)
1/2)/(2(a3b4 − b3a4)). (B.2)
Acknowledgements
We are thankful to V. Gorbounov for indicating us the strategy of the first proof of the tetra-
hedron equation in the trigonometric case in Section 4.2. The research was supported by the
Russian Science Foundation (project 20-61-46005). The authors thank the anonymous referees
for very useful comments which are improved the paper a lot.
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1 Introduction
1.1 Organization of the paper
2 Biggs interaction models
2.1 n-Potts models and Tutte polynomial
2.2 n-Potts models and the theorem of Matiyasevich
2.3 Shifting the order in the Potts models
3 Star-triangle equation for Ising and Potts models
3.1 General properties
3.2 The case n=2
3.3 The case n>2
4 Tetrahedron equation
4.1 Local Yang–Baxter equation
4.2 Tetrahedron equation, first proof
4.3 Tetrahedron equation, second proof
4.3.1 Involution lemma
4.3.2 Towards the tetrahedron equation
4.3.3 Solution for the tetrahedron equation
5 Star-triangle transformation, Biggs formula and conclusion
A The proof of Lemma 4.3
B For the second proof of the tetrahedron equation
References
|
| id | nasplib_isofts_kiev_ua-123456789-211314 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T06:05:41Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bychkov, Boris Kazakov, Anton Talalaev, Dmitry 2025-12-29T11:09:12Z 2021 Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. Boris Bychkov, Anton Kazakov and Dmitry Talalaev. SIGMA 17 (2021), 035, 30 pages 1815-0659 2020 Mathematics Subject Classification: 82B20; 16T25; 05C31 arXiv:2005.10288 https://nasplib.isofts.kiev.ua/handle/123456789/211314 https://doi.org/10.3842/SIGMA.2021.035 We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 2. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter n. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of = 2 multivariate Tutte polynomial. We extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute. We are thankful to V. Gorbounov for indicating to us the strategy of the first proof of the tetrahedron equation in the trigonometric case in Section 4.2. The research was supported by the Russian Science Foundation (project 20-61-46005). The authors thank the anonymous referees for their very useful comments, which have improved the paper a lot. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation Article published earlier |
| spellingShingle | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation Bychkov, Boris Kazakov, Anton Talalaev, Dmitry |
| title | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_full | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_fullStr | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_full_unstemmed | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_short | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_sort | functional relations on anisotropic potts models: from biggs formula to the tetrahedron equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211314 |
| work_keys_str_mv | AT bychkovboris functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation AT kazakovanton functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation AT talalaevdmitry functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation |