Legendrian DGA Representations and the Colored Kauffman Polynomial
For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combinati...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
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Інститут математики НАН України
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| Цитувати: | Legendrian DGA Representations and the Colored Kauffman Polynomial. Justin Murray and Dan Rutherford. SIGMA 16 (2020), 017, 33 pages |
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| author | Murray, Justin Rutherford, Dan |
| author_facet | Murray, Justin Rutherford, Dan |
| citation_txt | Legendrian DGA Representations and the Colored Kauffman Polynomial. Justin Murray and Dan Rutherford. SIGMA 16 (2020), 017, 33 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R¹ₙ, 𝛫(q) arises as a specialization 𝘍ₙ, 𝛫(a, q)∣ₐ⁻¹₌₀ of the n-colored Kauffman polynomial and (ii) when q is a power of two R¹ₙ, 𝛫(q) agrees with the total ungraded representation number, Rep₁(𝛫, 𝔽ⁿq), which is a normalized count of n-dimensional representations of (A(𝛫),∂) over the finite field 𝔽q. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m≠1.
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| first_indexed | 2025-12-17T12:04:17Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 017, 33 pages
Legendrian DGA Representations
and the Colored Kauffman Polynomial
Justin MURRAY † and Dan RUTHERFORD ‡
† Department of Mathematics, 303 Lockett Hall, Louisiana State University,
Baton Rouge, LA 70803-4918, USA
E-mail: jmurr24@lsu.edu
‡ Department of Mathematical Sciences, Ball State University,
2000 W. University Ave., Muncie, IN 47306, USA
E-mail: rutherford@bsu.edu
Received August 28, 2019, in final form March 10, 2020; Published online March 22, 2020
https://doi.org/10.3842/SIGMA.2020.017
Abstract. For any Legendrian knot K in standard contact R3 we relate counts of ungraded
(1-graded) representations of the Legendrian contact homology DG-algebra (A(K), ∂) with
the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ru-
ling polynomial, R1
n,K(q), as a linear combination of reduced ruling polynomials of positive
permutation braids and show that (i) R1
n,K(q) arises as a specialization Fn,K(a, q)
∣∣
a−1=0
of the n-colored Kauffman polynomial and (ii) when q is a power of two R1
n,K(q) agrees
with the total ungraded representation number, Rep1
(
K,Fn
q
)
, which is a normalized count
of n-dimensional representations of (A(K), ∂) over the finite field Fq. This complements
results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55–118] concerning
the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded
ruling polynomials with m 6= 1.
Key words: Legendrian knots; Kauffman polynomial; ruling polynomial; augmentations
2020 Mathematics Subject Classification: 53D42; 57M27
To Dmitry Fuchs on his 80th birthday with gratitude and admiration!
1 Introduction
The results of this article strengthen the connection between invariants of Legendrian knots in
standard contact R3 and the 2-variable Kauffman polynomial. Relations between the 2-variable
knot polynomials (HOMFLY-PT and Kauffman) and Legendrian knot theory were first realized
in the work of Fuchs and Tabachnikov [15] who observed, based on results of Bennequin [2],
Franks–Morton–Williams [12], and Rudolph [28], that these polynomials provide upper bounds
on the Thurston–Bennequin number of a Legendrian knot. At that time, it was still unknown
whether Legendrian knots in R3 were determined up to Legendrian isotopy by their Thurston–
Bennequin number, rotation number, and underlying smooth knot type (the so-called “classical
invariants” of Legendrian knots). This question was soon resolved with the introduction of
several non-classical invariants in the late 90’s and early 2000’s including the Legendrian contact
homology algebra which is a differential graded algebra (DGA) coming from J-holomorphic curve
theory that was constructed by Chekanov in [6] and discovered independently by Eliashberg
and Hofer, and combinatorial invariants introduced by Chekanov and Pushkar [7] defined by
This paper is a contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor
of Dmitry Fuchs. The full collection is available at https://www.emis.de/journals/SIGMA/Fuchs.html
mailto:jmurr24@lsu.edu
mailto:rutherford@bsu.edu
https://doi.org/10.3842/SIGMA.2020.017
https://www.emis.de/journals/SIGMA/Fuchs.html
2 J. Murray and D. Rutherford
counting certain decompositions of front diagrams called normal rulings.1 Interestingly, normal
rulings were discovered independently by Fuchs in connection with augmentations of the Le-
gendrian contact homology DGA. Moreover, Fuchs again pointed toward a connection between
Legendrian invariants and topological knot invariants by conjecturing in [13] that a Legendrian
knot should have a normal ruling if and only if the Kauffman polynomial estimate for the
Thurston–Bennequin number is sharp. This conjecture was resolved affirmatively in [29] by
interpreting Chekanov and Pushkar’s combinatorial invariants as polynomials, and showing that
the ungraded ruling polynomial, R1
K(z), of a Legendrian knot K ⊂ R3 arises as a specialization
R1
K(z) = FK(a, z)|a−1=0 (1.1)
of the framed version of the Kauffman polynomial FK ∈ Z
[
a±1, z±1
]
; the specialization has the
property that it is non-zero if and only if the Kauffman polynomial estimate for tb(K) is sharp.
An analogous result also established in [29] holds for the 2-graded ruling polynomial, R2
K(z),
and the HOMFLY-PT polynomial.
Initially, the Legendrian invariance of the ruling polynomials, based on establishing bijections
between rulings during bifurcations of the front diagram occurring in a Legendrian isotopy, was
somewhat mysterious from the point of view of symplectic topology. Building on the earlier
works [13, 14, 18, 26, 31], Henry and the second author showed in [19] that the ruling poly-
nomials are in fact determined by the Legendrian contact homology DGA, (A, ∂), since their
specializations at z = q1/2 − q−1/2 with q a prime power agree with normalized counts of aug-
mentations of (A, ∂) to the finite field Fq, i.e., DGA representations from (A, ∂) to (Fq, 0). Thus,
in the ungraded case (1.1) shows that counts of ungraded augmentations are actually topological
(depending only on the underlying framed knot type of K), as they arise from a specialization
of the Kauffman polynomial. In this article we extend this result by relating counts of higher
dimensional (ungraded) representations of (A, ∂) with the n-colored Kauffman polynomials.
To give a statement of our main result, for n ≥ 1, let Rep1
(
K,Fnq
)
denote the ungraded total
n-dimensional representation number of K as defined in [20]; see Definition 4.1. Let Fn,K(a, q)
denote the n-colored Kauffman polynomial (for framed knots); see Definition 3.2. In Section 2,
we define an ungraded n-colored ruling polynomial denoted R1
n,K(z).
Theorem 1.1. For any Legendrian knot K in R3 with its standard contact structure and any
n ≥ 1, there is a well-defined specialization Fn,K(a, q)|a−1=0, and we have
Rep1
(
K,Fnq
)
= R1
n,K(z) = Fn,K(a, q)|a−1=0.
As an immediate consequence, we get:
Corollary 1.2. The ungraded total n-dimensional representation number Rep1
(
K,Fnq
)
depends
only on the underlying framed knot type of K.
The corollary is a significant strengthening of a result from [24] that the existence of an
ungraded representation of (A, ∂) on Fn2 depends only on the Thurston–Bennequin number and
topological knot type of K. Precisely how much of the Legendrian contact homology DGA is
determined by the framed knot type of K remains an interesting question. See [24] and [22] for
some open conjectures along this line.
A previous article [20] establishes analogous results in the case of 2-graded representations
and the HOMFLY-PT polynomial, and in fact establishes the equality Repm
(
K,Fnq
)
= Rmn,K(z)
between the m-graded total representation numbers and m-graded colored ruling polynomials
for all m ∈ Z≥0 except for the ungraded case where m = 1. The m = 1 case is more involved for
1Around the same time, generating family homology invariants capable of distinguishing Legendrian links with
the same classical invariants were introduced by Traynor [33].
Legendrian DGA Representations and the Colored Kauffman Polynomial 3
a number of reasons. In the following we briefly review the argument from [20] and then outline
our approach to Theorem 1.1.
For m 6= 1, the n-colored ruling polynomial is defined as a linear combination of satellite
ruling polynomials of the form
Rmn,K(q) =
1
cn
∑
β∈Sn
qλ(β)/2RmS(K,β)(z)
∣∣
z=q1/2−q−1/2 , m 6= 1, (1.2)
where S(K,β) is the Legendrian satellite of K with a Legendrian positive permutation braid
associated to β ∈ Sn. The same linear combination of HOMFLY-PT polynomials defines the
n-colored HOMFLY-PT polynomial. In [20], the total n-dimensional representation number
is recovered from (1.2) via a bijection between m-graded augmentations of S(K,β) and n-
dimensional representations of the DGA of K mapping a distinguished invertible generator into
Bβ ⊂ GL(n,F) where GL(n,F) = tβ∈SnBβ is the Bruhat decomposition. Thus, summing
over all β ∈ Sn corresponds to considering all n-dimensional representations of the DGA of K
on Fn.
When m = 1, the above bijection becomes modified in an interesting way, as augmenta-
tions of S(K,β) now correspond to (ungraded) representations of (A, ∂) on differential vector
spaces of the form
(
Fn, d
)
where d varies over all (ungraded) upper triangular differential on Fn.
(When m 6= 1, d = 0 is automatic for grading reasons.) The total n-dimensional representation
number, Rep1
(
K,Fnq
)
, only counts representations with d = 0, so the definition of R1
n,K needs
to be changed to only take into account normal rulings corresponding to representations with
d = 0. This is done by replacing each R1
S(K,β) in (1.2) with the corresponding reduced ruling
polynomial R̃1
S(K,β) as introduced in [24] that only counts normal rulings of S(K,β) that never
pair two strands of the satellite that correspond to a single strand of K. Up to a technical point
about the use of different diagrams in [20] and [19] that the bulk of Section 4 is spent addressing,
this leads to the equality Rep1
(
K,Fnq
)
= R1
n,K(z).
Establishing that R1
n,K(z) = Fn,K(a, q)|a−1=0 requires a much more involved argument than
for the case of the colored HOMFLY-PT polynomial and R2
n,K(z) (where the result is immediate
from [29] and the definition). The n-colored Kauffman polynomial is defined by satelliting K
with the symmetrizer in the BMW algebra, Yn ∈ BMWn. In addition to a sum over permutation
braids as in the HOMFLY-PT case, Yn also has terms of a less explicit nature (though, see [8])
involving tangle diagrams in [0, 1]×R with turn-backs, i.e., components that have both endpoints
on the same boundary component of [0, 1]×R. To relate R1
n,K and Fn,K , we use the combinatorics
of normal rulings to find an inductive characterization of R1
n,K in terms of ordinary ruling
polynomials rather than reduced ruling polynomials, and then compare this with an inductive
characterization of Yn due to Heckenberger and Schüler [17].
The remainder of the article is organized as follows. In Section 2, we define the ungraded
n-colored ruling polynomial and establish an inductive characterization of it in Theorem 2.8.
In Section 3, we recall the definition of the colored Kauffman polynomial and prove the second
equality of Theorem 1.1 (see Theorem 3.9). Section 4 reviews definitions of representation
numbers from [20] and then establishes the first equality of Theorem 1.1 (see Theorem 4.2). In
Section 5, we close the article with a brief discussion of a modification of Theorem 1.1 for the
case of multi-component Legendrian links.
2 The n-colored ungraded ruling polynomial
In this section, after a brief review of ruling polynomials and Legendrian satellites, we define the
ungraded n-colored ruling polynomial R1
n,K as a linear combination of reduced ruling polynomials
indexed by permutations β ∈ Sn. Reduced rulings, considered earlier in [24], form a restricted
4 J. Murray and D. Rutherford
Figure 1. Each closed curve of a normal ruling consists of a pair of companion paths with monotonically
increasing x-coordinate beginning and ending at a common left and right cusp of πxz(L). At switches,
paths from two different closed curves of ρ meet and both turn a corner at a crossing. The normality
condition requires that near switches the switching paths and their companion paths match one of the
pictured configurations. At crossings that are not switches paths from two different closed curves cross
transversally.
class of normal rulings of satellite links, so that it is not immediately clear how to describe R1
n,K
in terms of ordinary ruling polynomials. For this purpose, we work in a Legendrian version of
the n-stranded BMW algebra, BMWLeg
n , and inductively construct elements Ln ∈ BMWLeg
n that
can be used to produce R1
n,K via (non-reduced) satellite ruling polynomials.
2.1 Legendrian fronts and ruling polynomials
In this article we consider Legendrian links and tangles in a 1-jet space J1M where M is one
of R, S1, or [0, 1]. In all cases, we can view J1M = T ∗M ×R as M ×R2, and using coordinates
(x, y, z) with x ∈ M and y, z ∈ R the contact form is dz − y dx. Legendrian curves can be
viewed via their front projection πxz : J1M → M × R, (x, y, z) 7→ (x, z) which is a collection of
curves having cusp singularities and transverse double points but no vertical tangencies. The
original Legendrian is recovered via y = dz
dx , so in front diagrams (implicitly) the over-strand
at a crossing is the strand with lesser slope (as the y-axis is oriented away from the viewer).
Legendrian links have a contact framing which is the framing given by the upward unit normal
vector to the contact planes.
− = z
(
−
)
, (R1)
= = 0, (R2)
tK = z−1K. (R3)
Figure 2. The ungraded ruling polynomial skein relations.
Recall that for a Legendrian link K ⊂ J1R a normal ruling ρ of K is a decomposition of the
front diagram of K into a collection of simple closed curves with corners at a left and right cusp
and at switches (adhering to the normality condition, see Fig. 1). For each x = x0 where the
front projection of K does not have crossings or cusps, a normal ruling ρ divides the strands
of K at x = x0 into pairs, so that ρ can be viewed as a sequence of pairings of strands of K.
For a more detailed discussion of normal rulings see for instance [13, 24, 29, 31]. The ungraded
ruling polynomial of L (also called the 1-graded ruling polynomial) is defined as
R1
K(z) :=
∑
ρ∈Γ(K)
zj(ρ) ∈ Z
[
z±1
]
,
Legendrian DGA Representations and the Colored Kauffman Polynomial 5
where the sum is over all normal rulings of K and j(ρ) = #switches−#right cusps. For Legen-
drian links in J1R, the ungraded ruling polynomial of K satisfies and is uniquely characterized
by the skein relations in Fig. 2 and the normalization R1 = z−1. (See [29].) The relations in
Fig. 2 imply two additional relations that we will make use of, cf. [30, Section 6].
fishtail relation: = = 0, (2.1)
double-crossing relation: = + z · − z · .
Remark 2.1. The double crossing relation can be used to show that the first relation of Fig. 2
also hold with right cusps. Moreover, the third relation from Fig. 2 is implied by the first two
as long as K 6= ∅.
2.2 A Legendrian BMW algebra
A Legendrian n-tangle is a properly embedded Legendrian α ⊂ J1[0, 1] (i.e., compact with
∂α ⊂ ∂J1[0, 1]) whose front projection agrees with the collection of horizontal lines z = i,
1 ≤ i ≤ n, near ∂J1[0, 1]. Legendrian isotopies of n-tangles are required to remain fixed in
a neighborhood of ∂J1[0, 1]. At x = 0 and x = 1 we enumerate the endpoints and strands of
a Legendrian tangle from 1 to n with descending z-coordinate. For any permutation β ∈ Sn,
there is a corresponding positive permutation braid that is a Legendrian n-tangle, also denoted
β ⊂ J1[0, 1], that connects endpoint i at x = 0 to endpoint β(i) at x = 1 for 1 ≤ i ≤ n. Up to
Legendrian isotopy, β is uniquely characterized by requiring that
(i) the front projection does not have cusps and,
(ii) for i < j, the front projection has no crossings (resp. exactly one crossing) between the
strands with endpoints i and j at x = 0 if β(i) < β(j) (resp. β(i) > β(j)).
The number of crossings in such a front diagram for β is called the length of β and will be deno-
ted λ(β). For Legendrian n-tangles α, β ⊂ J1[0, 1], we define their multiplication α ·β ⊂ J1[0, 1]
by stacking β to the left of α (as in composition of permutations). Diagrammatically:
α · β = β α
Definition 2.2. LetR be a coefficient ring containing Z[z±1] as a subring. Define the Legendrian
BMW algebra, BMWLeg
n , as an R-module to be the quotient RLegn/S where RLegn is the free
R-module generated by Legendrian isotopy classes of Legendrian n-tangles in J1[0, 1], and S is
the R-submodule generated by the ruling polynomial skein relations from Fig. 2. Multiplication
of n-tangles induces an R-bilinear product on BMWLeg
n .
In the remainder of the article, we fix the coefficient ring R to be Z
[
s±1
]
localized to include
denominators of the form sn− s−n for n ≥ 1 where z = s− s−1. In Section 4, we will work with
the alternate variable s = q1/2.
Fig. 3 indicates crossing and hook elements, σi, ei ∈ BMWLeg
n , for 1 ≤ i < n. Note that the
fishtail and double-crossing relations imply
σiei = eiσi = 0, and (2.2)
σ2
i = 1 + zσi − zei, for 1 ≤ i < n. (2.3)
6 J. Murray and D. Rutherford
σi := ei :=
Figure 3. Crossing and hook elements in BMWLeg
n .
K
β
S(K,β)
Figure 4. The Legendrian satellite S(K,β) where β = σ1σ2 ⊂ J1[0, 1] is the positive permutation braid
associated to the 3-cycle (1 2 3). The J1[0, 1]-part of the satellite is indicated by the dotted rectangular
box.
Remark 2.3.
1. Occasionally, we will multiply an element α ∈ BMWLeg
i by an element of BMWLeg
j with
i < j. Unless indicated otherwise, we do so by extending i-tangles to j-tangles by placing
j − i horizontal strands below α.
2. We do not give a complete set of generators and relations for BMWLeg
n , as the main role of
BMWLeg
n in this article is to provide a convenient setting for ruling polynomial calculations.
We leave finding a presentation for BMWLeg
n as an open problem.
2.3 Legendrian satellites and reduced ruling polynomials
We will be considering rulings of satellites. Given a connected Legendrian knot K ⊂ J1R and
a Legendrian n-tangle L ⊂ J1[0, 1], a front diagrammatic description of the Legendrian satellite
S(K,L) ⊂ J1R is the following: Form the n-copy of K, i.e., take n copies of K each shifted
up a small amount in the z-direction, then insert a rescaled version of the n-tangle L into the
n-copy at a small rectangular neighborhood J ∼= [0, 1]× [−ε, ε] of part of a strand of K that is
oriented from left-to-right. We refer to J as the J1[0, 1]-part of the satellite. See Fig. 4. It can
be shown that the Legendrian isotopy type of S(K,L) depends only on K and the closure of L
in J1S1, cf. [27]. (See also Section 4.4 for additional discussion.)
In [24] a variant of the ruling polynomial was introduced for satellites using reduced normal
rulings. Let L ⊂ J1[0, 1] be a Legendrian n-tangle. A normal ruling ρ of S(K,L) is said to be
reduced if, outside of the J1[0, 1]-part of S(K,L), parallel strands of S(K,L) corresponding to
a single strand of K are not paired by ρ. See Fig. 5. We denote the set of reduced normal rulings
of S(K,L) by Γ̃(K,L) and define the reduced ruling polynomial of S(K,L) as R̃S(K,L)(z) :=∑
ρ∈Γ̃(K,L)
zj(ρ).
Legendrian DGA Representations and the Colored Kauffman Polynomial 7
Figure 5. A reduced normal ruling (left) and a non-reduced normal ruling (right) of S(K,β) where K
is a right-handed trefoil and β = 1.
Remark 2.4. If the reduced condition holds for a normal ruling ρ of S(K,L) for the parallel
strands of S(K,L) corresponding to a single point k0 ∈ K outside of the J1[0, 1]-part of S(K,L),
then ρ will be reduced. Indeed, the involution of parallel strands of S(K,L) at k0 ∈ K that
arises from restriction the pairing of ρ (strands of S(K,L) at k0 that are paired with strands in
a part of the satellite away from k0 are fixed points of the involution) is called the “thin part”
of ρ at k0 in [24], and Lemmas 3.3 and 3.4 from [24] show that the thin part is independent
of k0.
We are now prepared to make the following key definition.
Definition 2.5. For any (connected) Legendrian knot K ⊂ J1R we define the ungraded n-
colored ruling polynomial as
R1
n,K(s) :=
1
cn
∑
β∈Sn
sλ(β)R̃S(K,β)(z),
where the sum is over positive permutation braids, z = s − s−1, λ(β) is the length of β, and
cn = sn(n−1)/2
n∏
i=1
si−s−i
s−s−1 .
Note that R1
n,K belongs to coefficient ring R defined above.
Remark 2.6. It is proved in [24] that with L fixed the reduced ruling polynomial R̃S(K,L)(z) is
a Legendrian isotopy invariant of K. Alternatively, the Legendrian isotopy invariance of R1
n,K
also follows from Theorem 2.8 below and invariance of the ordinary ruling polynomials.
2.4 Inductive characterization of R1
n,K
It is convenient to extend the concept of satellite ruling polynomials slightly by defining R1
S(K,η)
for η ∈ BMWLeg
n represented as an R-linear combination of n-tangles
k∑
i=1
riαi to be
R1
S(K,η) =
k∑
i=1
riR
1
S(K,αi)
. (2.4)
(This is well-defined since the front diagram of each αi appears as a subset of πxz(S(K,αi))
and R1 satisfies the skein relations that define BMWLeg
n .) The goal in the remainder of this
section is to give such a characterization of the ungraded n-colored ruling polynomials via
appropriate elements Ln ∈ BMWLeg
n .
8 J. Murray and D. Rutherford
Let
γn = 1 +
n−1∑
j=1
sjσn−1σn−2 · · ·σn−j ∈ BMWLeg
n ,
where σi is as in Fig. 3 and consider the elements of BMWLeg
n defined inductively for n ≥ 1 by
L1 = 1 and
Ln = Ln−1βn, n ≥ 2,
where
βn :=
(
1− z
n∑
k=2
αk,n
)
γn.
Here, αk,n ∈ BMWLeg
n is defined inductively on the bottom k strands by α2,n = en−1 and
αk,n = − z · − z · + z2 · (2.5)
with the identity tangle appearing as the top n − k strands of αk,n. For example, using (2.2)
we have L2 = 1 + sσ1 − ze1. The double subscript notation on αk,n is to emphasize that αk,n
involves the bottom k strands, n− k + 1, n− k + 2, . . . , n, out of n total strands. (By contrast,
when multiplying Ln−1βn an extra strand is placed at the bottom of Ln−1; see Remark 2.3.)
In Section 3, we will also make use of a non-inductive characterization of the αk,n. Given
a front diagram D, let cr(D) denote the set of crossings appearing in D. We define the resolution
of D with respect to X ⊂ cr(D) to be the tangle rX(D) obtained from resolving all crossings
of X as depicted in Fig. 6.
−→
Figure 6. The resolution of a crossing in X ⊂ cr(D).
Lemma 2.7. For 1 < k ≤ n,
αk,n =
∑
X⊂cr(Ck,n)
(−z)|X|rX(Ck,n), where Ck,n =
Proof. This is a straightforward induction on k. �
Theorem 2.8. For any Legendrian K ⊂ J1R, R1
n,K = 1
cn
R1
S(K,Ln).
The proof will be given below after Lemma 2.10.
Given K ⊂ J1R, and η a Legendrian n-tangle, recall that the J1[0, 1]-part of S(K, η) refers
to a rectangular region, J ∼= [0, 1] × [−ε, ε], where the front diagram of η appears. We will call
a normal ruling ρ of S(K, η) k-reduced if at the right boundary of J none of the top k parallel
Legendrian DGA Representations and the Colored Kauffman Polynomial 9
strands of η within S(K, η) are paired with one another. We denote by Γ̃k(K, η), the set of
k-reduced normal rulings of S(K, η). Similarly, given a location ∗ within J corresponding to
an x-value in J1[0, 1] without double points of η a normal ruling ρ of S(K, η) is said to be
(k,m)-paired at ∗ if ρ is (m− 1)-reduced (at the right boundary of J) and ρ pairs strand k of η
with strand m of η at ∗ (where strands are numbered from top to bottom as they appear in η
at ∗). The set of (k,m)-paired rulings of S(K, η) is denoted by Γ
τ(k,m)
∗ (K, η). The k-reduced
ruling polynomial and (k,m)-paired ruling polynomial (at ∗) are defined respectively by
R̃
(k)
(K,η)(z) =
∑
ρ∈Γ̃k(K,η)
zj(ρ) and R
τ(k,m)
(∗;K,η)(z) =
∑
ρ∈Γ
τ(k,m)
∗ (K,η)
zj(ρ).
Either polynomial can be extended by linearity to allow η to be an R-linear combination of
Legendrian n-tangles.
Remark 2.9. If η is a n-stranded positive braid, then any n-reduced normal ruling is reduced.
This is because if parallel strands of the satellite S(K, η) corresponding to a single strand of K
are not paired at the right side of the J1[0, 1]-part of the satellite, then such parallel strands
cannot be paired anywhere outside of the J1[0, 1]-part. See Remark 2.4.
Lemma 2.10. For any 1 < k ≤ m ≤ n, let β ∈ Sm−1 be a positive permutation braid extended
to an Legendrian n-tangle by placing the n −m + 1 stranded identity tangle below β, and let ν
be a linear combination of Legendrian n-tangles. Then, we have
zR̃
(m−1)
(K,βαk,mν)(z) = R
τ(m−k+1,m)
(∗;K,βν) (z),
where ∗ is the location between β and ν (immediately to the left diagrammatically of β).
Proof. The proof is by induction on k with m and n fixed. The base case is clear since
α2,m = em−1.
Suppose the statement is true for k. For clarity of this proof, we emphasize the location of
our paired rulings within our notation, and abbreviate αk,m as αk. Using (2.5) and the inductive
hypothesis, we compute
zR̃
(m−1)
(K,βαk+1ν)(z)
= zR̃
(m−1)
(K,βσm−kαkσm−kν)(z)− z
2R̃
(m−1)
(K,βσm−kαkν)(z)− z
2R̃
(m−1)
(K,βαkσm−kν)(z) + z3R̃
(m−1)
(K,βαkν)(z)
= R
τ(m−k+1,m)
(∗;K,βσm−k∗σm−kν)(z)− zR
τ(m−k+1,m)
(∗;K,βσm−k∗ν)(z)− zR
τ(m−k+1,m)
(∗;K,β∗σm−kν)(z) + z2R
τ(m−k+1,m)
(∗;K,β∗ν) (z)
?
= R
τ(m−k,m)
(∗;K,β∗ν) (z).
We now establish equality ?. For the diagrams D that follow, when considering (k,m)-paired
rulings, we indicate the location ∗ with a green vertical segment having endpoints on the paired
strands k and m. When we encode such information in D we suppress the paired notation
R
τ(k,m)
(∗;K,D) as RτK,D (and similarly for sets of rulings: Γ
τ(k,m)
∗ (K,D) becomes Γτ (K,D)). For
instance,
R
τ(m−k,m)
(∗;K,β∗ν) (z) = Rτ(
K,
)(z),
where the diagram only pictures strands with numberings in the range m−k to m at ∗. Observe
the bijection
Γτ
(
K,
)
=
{
ρ ∈ Γτ
(
K,
) ∣∣∣∣ r1 and r2 are not switches of ρ
}
, (2.6)
10 J. Murray and D. Rutherford
where r1 and r2 denote the left and right crossings of the diagram on the right. In addition, we
have the following (separate) bijections:
ρ 7→ Φ(ρ),{
ρ ∈ Γτ
(
K,
) ∣∣ r1 is a switch
}
←→ Γτ
(
K,
)
,{
ρ ∈ Γτ
(
K,
) ∣∣ r2 is a switch
}
←→ Γτ
(
K,
)
,{
ρ ∈ Γτ
(
K,
) ∣∣ r1 and r2 are switches
}
←→ Γτ
(
K,
)
,
(2.7)
where Φ(ρ) is the unique ruling that agrees with ρ outside of a neighborhood of the resolved
crossing(s). (Any ruling of a diagram D with switches at a set of crossings X gives rise to a ruling
of rX(D).) Each Φ is clearly injective. To verify surjectivity, observe that any ruling ρ′ in one
of the sets on the right side of (2.7) indeed comes from a ruling on the left side by replacing
the resolved crossing(s) with switches. Here, it is crucial to note that the normality condition
is automatically satisfied. If the normality condition was not satisfied, then ρ′ must pair strand
m−k with a strand numbered in the range m−k+2 to m−1 at the location ∗ indicated by the
vertical segment. Since β ∈ Sm−1, it follows that at least 2 of the top m− 1 strands are paired
at the right side of the J1[0, 1]-part of the satellite, contradicting that ρ′ is (m − 1)-reduced.
Finally, the inclusion-exclusion principle and the above bijections (2.6) and (2.7) imply
Rτ(
K,
)(z)
= Rτ(
K,
)(z)− zRτ(
K,
)(z)− zRτ(
K,
)(z) + z2Rτ(
K,
)(z).
(The factors of z and z2 appear since the first two bijections in (2.7) decrease the number of
switches by 1, while the last bijection decreases the number by 2.) This establishes ?, so we are
done. �
Proof of Theorem 2.8. With n ≥ 1 fixed we prove the following statement by induction on m.
The theorem follows from the special case where m = n and µ = 1 (in view of Remark 2.9).
Inductive statement: For any (linear combination of) Legendrian n-tangle(s) µ ⊂ J1[0, 1]
and any 1 ≤ m ≤ n,
RS(K,Lmµ)(z) =
∑
β∈Sm
sλ(β)R̃
(m)
(K,βµ)(z)
(with the products Lmµ and βµ formed as in Remark 2.3).
The base case of m = 1 is immediate since any normal ruling is 1-reduced. For the inductive
step, with m ≥ 2 and the statement assumed for m− 1 we compute
RS(K,Lmµ)(z) = R
S(K,Lm−1(1−z
m∑
k=2
αk,m)γmµ)
(z)
= RS(K,Lm−1γmµ) − z
m∑
k=2
RS(K,Lm−1αk,mγmµ)(z)
=
∑
β∈Sm−1
sλ(β)
(
R̃
(m−1)
(K,βγmµ)(z)− z
m∑
k=2
R̃
(m−1)
(K,βαk,mγmµ)(z)
)
(inductive hyp)
=
∑
β∈Sm−1
sλ(β)
(
R̃
(m−1)
(K,βγmµ)(z)−
m∑
k=2
R
τ(m−k+1,m)
(∗;K,β∗γmµ) (z)
)
. (Lemma 2.10)
Legendrian DGA Representations and the Colored Kauffman Polynomial 11
Note that since there are no crossings in β involving the m-th strand, an (m− 1)-reduced ruling
is m-reduced unless at ∗ strand m is paired with one of the strands numbered 1, . . . ,m − 1 in
the J1[0, 1]-part of the satellite, i.e., a strand numbered m − k + 1 with 2 ≤ k ≤ m. Thus,
continuing the above computation, we have
∑
β∈Sm−1
sλ(β)
(
R̃
(m−1)
(K,βγmµ)(z)−
m∑
k=2
R
τ(m−k+1,m)
(∗;K,β∗γmµ) (z)
)
=
∑
β∈Sm−1
sλ(β)R̃
(m)
(K,βγmµ).
Next note that any β ∈ Sm can be written uniquely as β = β′σm−1 · · ·σm−j with β′ ∈ Sm−1 and
0 ≤ j ≤ m− 1. (Here, m− j = β−1(m).) Moreover, such a factorization realizes β as a positive
permutation braid, so that λ(β) = λ(β′) + j. Using the definition of γm, we see that∑
β∈Sm−1
sλ(β)R̃
(m)
(K,βγmµ) =
∑
β∈Sm
sλ(β)R̃
(m)
(K,βµ).
This completes the inductive step. �
3 Relation to the Kauffman polynomial
We begin this section with a review of the definition of the n-colored Kauffman polynomial Fn,K
via satelliting with the symmetrizer Yn ∈ BMWn in the BMW algebra. Then, using an in-
ductive characterization of Yn from [17], combined with the earlier characterization of R1
n,K we
show how to recover R1
n,K as a specialization Fn,K
∣∣
a−1=0
. This is accomplished by relating the
Legendrian BMW element Ln ∈ BMWLeg
n from Theorem 2.8 to Yn in a certain sub-quotient
of BMWn defined in terms of Legendrian tangles. Along the way we pause to make a conjecture
relating ungraded ruling polynomial skein modules with Kauffman skein modules in general.
3.1 The Kauffman polynomial and the BMW algebra
Recall that the (framed) Kauffman polynomial (Dubrovnik version) is a regular isotopy invariant
that assigns a Laurent polynomial to framed links L ⊂ R3 characterized by the skein relations
(shown with blackboard framing)
− = z
−
, (F1)
= a−1 = a , (F2)
t L =
(
a− a−1
z
+ 1
)
L, (F3)
where FL := 1 if L is the empty link.2
The ordinary BMW algebra, BMWn (see [3, 21]) can be defined as the Kauffman skein
module for framed tangles in [0, 1]× R2 ∼= J1[0, 1] with n boundary points on each component.
In more detail, let Frn denote the set of isotopy classes of framed n-tangles where we require
that near ∂J1[0, 1] framed n-tangles agree with the horizontal lines z = i, y = 0, 1 ≤ i ≤ n with
framing vector ∂
∂y ; this requirement is maintained during isotopies. Working over the field of
2This is equivalent to choosing the normalization FU = a−a−1
z
+ 1, where U is the 0-framed unknot.
12 J. Murray and D. Rutherford
L fr(L)
Figure 7. The framed n-tangle fr(L) associated to a Legendrian n-tangle L ⊂ J1[0, 1].
rational functions F = Z(a, s) with z = s− s−1, BMWn = FFrn/T where T is the F-submodule
generated by the Kauffman polynomial skein relations. Multiplication in BMWn is as in the
Legendrian case, i.e., the product α · β is β stacked to the left of α.
For viewing framed n-tangles diagrammatically, we continue to use the xz-plane for projec-
tions, and we require the framing to be globally given by ∂
∂y , that is, perpendicular to the projec-
tion plane and away from the viewer. This framing becomes isotopic to the blackboard framing
when n-tangles are closed to become links in J1S1 (by identifying the left and right boundaries
of J1[0, 1]). To view a Legendrian n-tangle L ⊂ J1[0, 1] as a framed n-tangle fr(L) ∈ Frn we
form a diagram for fr(L) from the front diagram of L by smoothing left cusps and adding a small
loop with a negative crossing at right cusps. See Fig. 7. This has the property that the closure
of fr(L) in J1S1 is framed isotopic to the Legendrian closure of L with its contact framing.
The topological satellite operation produces from a framed knot K ⊂ J1R and a framed
n-tangle L ⊂ J1[0, 1] a satellite link S(K,L) ⊂ J1R. A diagram for S(K,L) arises from taking
a blackboard framed diagram for K and placing a blackboard framed diagram for the closure
of L in an (immersed) annular neighborhood of K in the projection plane. When K and L
are Legendrian, the framed knot type of S(K,L) agrees with that of the previously defined
Legendrian satellite (with contact framing). For K ⊂ J1R (a framed link) and η ∈ BMWn, the
Kauffman polynomial of the satellite FS(K,η) is defined by linearity as in (2.4).
3.2 Symmetrizer in BMWn and the n-colored Kauffman polynomial
The n-colored Kauffman polynomial is defined by satelliting with the symmetrizer Yn ∈ BMWn.
(More general, colored Kauffman polynomials, where the coloring is by a partition λ, can be
defined using other idempotents in BMWn and are related to quantum invariants of type B, C,
and D; see, e.g., [1].) The symmetrizer Yn is characterized as the unique non-zero element of
BMWn that is idempotent, i.e., has Y2
n = Yn, and satisfies
Ynσi = sYn, for 1 ≤ i < n. (crossing absorbing property) (3.1)
The following inductive formula for Yn is due to Heckenberger and Schüler in [17]. Consider
Yn ∈ BMWn defined by
Y1 = 1,
Yn = Yn−1γn + Yn−1
z
1− s2n−3a
n−1∑
i=1
as2n−2i−1
i−1∑
j=0
sjDi−j,i+1
, (3.2)
Legendrian DGA Representations and the Colored Kauffman Polynomial 13
where
Di,j :=
,
i < j,
0, i ≥ j.
The symmetrizer is then obtained as the normalization Yn = Yn
cn
(with cn as in Definition 2.5).
For example, Y2 = 1
c2
(
id + sσ1 + sza
1−sae1
)
. Note that Yn is quasi-idempotent (i.e., Y 2
n = cnYn)
and also has the crossing absorbing property (3.1).
Remark 3.1. Our notations σi, s, a, ei translate into the notations from [17] as gi, q, r, r
−1ei.
(The r−1 is because our ei is a Legendrian hook, so that fr(ei) has an extra loop that produces the
r−1 factor.) Our Di,j is such that a·Di−j,i+1 is the j-th term in d+
n,i from [17], the factor of a again
arising from our use of Legendrian tangles in representing the quasi-idempotent Yn. Note that
[17, Proposition 1] gives an inductive formula for Yn (notated there as Sn) rather than Yn, and
the denominator qn[[n]] that appears there is accounted for by our factor of 1
cn
relating Yn and Yn.
Definition 3.2. Given a framed knot K ⊂ R3, we define the n-colored Kauffman polynomial
of K by
Fn,K(a, s) = FS(K,Yn)(a, z)|z=s−s−1 ,
where Yn is the symmetrizer in BMWn.
3.3 Ruling polynomials via specializations of the Kauffman polynomial
For a Legendrian link in L ⊂ J1R the Thurston–Bennequin number satisfies the inequal-
ity tb(L) ≤ − dega F̂L(a, z), where F̂L = a−w(L)FL is the framing independent version of
the Kauffman polynomial obtained by normalizing FL using the writhe of a diagram for L.
(See [15, 23, 32].) This is equivalent to the inequality
dega FL(a, z) ≤ 0. (3.3)
Thus, when L is Legendrian FL ∈ Z
[
a±1, z±
]
does not contain positive powers of a, and a spe-
cialization FL(a, z)|a−1=0 arises from simply setting a−1 = 0.
Theorem 3.3 ([29]). For any Legendrian link L ⊂ J1R, the ungraded ruling polynomial is the
specialization
R1
L(z) = FL(a, z)|a−1=0.
We will want to perform a similar specialization on elements of BMWn whose coefficients in
F = Z(a, s) may not be Laurent polynomials in a. To do so, note that the notion of degree in a
extends to arbitrary non-zero rational functions via dega : Z(a, s) \ {0} → Z,
dega
(
f
g
)
= dega f − dega g, for f, g ∈ Z[a, s].
In addition, we use the convention that dega 0 := −∞. Moreover, on the subring F− := {F ∈ F |
dega F ≤ 0} we can define a specialization |a−1=0 : F− → Z(s), by
f
g
∣∣∣∣
a−1=0
:=
0, dega(f/g) < 0,
ca(f)
ca(g)
, dega(f/g) = 0,
where ca(f) ∈ Z[s] denotes the leading coefficient in a of f .
14 J. Murray and D. Rutherford
Proposition 3.4. The specialization |a−1=0 : F− → Z(s) is a well-defined, unital ring homo-
morphism.
Proof. Suppose f1
g1
= f2
g2
in F−. Then f1g2 = g1f2 and so ca(f1)ca(g2) = ca(g1)ca(f2), since
ca(fg) = ca(f)ca(g) is true for polynomials f and g. It follows that f1
g1
∣∣
a−1=0
= f2
g2
∣∣
a−1=0
and
so the specialization
∣∣
a−1=0
is well-defined. Note that 1|a−1=0 = 1 follows from the definition.
Now suppose f1
g1
, f2g2 ∈ F−. If dega(f1/g1) = dega(f2/g2) = 0, then dega(f1g2) = dega(g1f2) =
dega(g1g2). In which case,(
f1
g2
+
f2
g2
) ∣∣∣∣
a−1=0
=
f1g2 + g1f2
g1g2
∣∣∣∣
a−1=0
=
ca(f1)ca(g2) + ca(g1)ca(f2)
ca(g1)ca(g2)
=
f1
g1
∣∣∣∣
a−1=0
+
f2
g2
∣∣∣∣
a−1=0
.
If dega(f1/g1),dega(f2/g2) < 0, then
dega(f1g2 + g1f2) ≤ max{dega(f1g2),dega(g1f2)} < dega(g1g2)
and so
(f1
g1
+ f2
g2
)∣∣
a−1=0
= 0 = f1
g1
∣∣
a−1=0
+ f2
g2
∣∣
a−1=0
holds by definition. Lastly, if dega(f1/g1) <
dega(f2/g2) = 0, then dega(f1g2 + g1f2) = dega(g1f2) = deg(g1g2) and(
f1
g2
+
f2
g2
) ∣∣∣∣
a−1=0
=
f1g2 + g1f2
g1g2
∣∣∣∣
a−1=0
=
ca(g1)ca(f2)
ca(g1)ca(g2)
=
f1
g1
∣∣∣∣
a−1=0
+
f2
g2
∣∣∣∣
a−1=0
.
It remains to show |a−1=0 preserves multiplication. In the case where dega(f1/g1) < 0, it follows
that dega(f1f2) < dega(g1g2) and so(
f1f2
g1g2
) ∣∣∣∣
a−1=0
= 0 =
f1
g1
∣∣∣∣
a−1=0
· f2
g2
∣∣∣∣
a−1=0
.
The remaining case where dega(f1/g1) = dega(f1/g1) = 0 immediately follows from the fact
that ca preserves multiplication for then(
f1f2
g1g2
) ∣∣∣∣
a−1=0
=
ca(f1f2)
ca(g1g2)
=
ca(f1)ca(f2)
ca(g1)ca(g2)
=
f1
g1
∣∣∣∣
a−1=0
· f2
g2
∣∣∣∣
a−1=0
. �
In the following we will make a connection between the Legendrian BMW algebra from
Section 2 and BMWn. Towards this end, define
BMW−
n = SpanF− Legn ⊂ BMWn
to be the F−-submodule generated by (framed isotopy classes of) Legendrian n-tangles. Next,
let m ⊂ F− denote the maximal ideal m = {F ∈ F− | dega F < 0}, that is the kernel of |a−1=0,
and consider the quotient F−-module
BMW∞
n = BMW−
n /m · BMW−
n ,
with the projection map notated as
BMW−
n → BMW∞
n , y 7→ y|a−1=0.
Note that the estimate (3.3) shows that for any fixed Legendrian knot K ⊂ J1R the F-module
homomorphism BMWn → F, L 7→ FS(K,L)(a, s) maps BMW−
n to F−. Moreover, the specializa-
tion BMW−
n → Z(s), L 7→ FS(K,L)(a, s)
∣∣
a−1=0
induces a well defined map BMW∞
n → Z(s).
Legendrian DGA Representations and the Colored Kauffman Polynomial 15
Proposition 3.5. There is an R-algebra homomorphism ϕ : BMWLeg
n → BMW∞
n induced by
the map Legn → Frn. Moreover, we have a commutative diagram
BMW−
n
FS(K,·) //
|a−1=0
��
F−
|a−1=0 // Z(s).
BMW∞
n
33
BMWLeg
n
ϕ
OO R1
S(K,·)
77
Proof. To see that ϕ is well-defined, note that the ruling polynomial relation (R1) holds in
BMWn since it is implied by the Kauffman relation (F1). Moreover, using the (F2), we see
that when L ⊂ J1[0, 1] has a zig-zag, and L′ ⊂ J1[0, 1] is the Legendrian obtained by removing
the zig-zag from L, we have L = a−1L′ in BMW−, and this implies that [L] = 0 in BMW∞ as
required by (R2). Finally, to verify (R3) when K ∈ Legn we can compute in BMW∞
n
K t = a−1K t = a−1
(
a− a−1
z
+ 1
)
K
= z−1K +
(
a−1 − a−2/z
)
K = z−1K,
since
(
a−1 − a−2/z
)
∈ m.
The upper triangle of the diagram is commutative by definitions, and the commutativity of
the lower triangle follows from Theorem 3.3. �
Conjecture 3.6. When BMWLeg
n is defined over Z(s), the map ϕ is an algebra isomorphism.
Remark 3.7. A similar conjecture can be made involving the (suitably defined) 1-graded ruling
polynomial skein module and the Kauffman skein module of any contact 3-manifold, M .
Recall the element Ln ∈ BMWLeg
n from Section 2.4.
Proposition 3.8. For any n ≥ 1, we have
1) Yn ∈ BMW−
n , and
2) ϕ
(
1
cn
Ln
)
= Yn
∣∣
a−1=0
holds in BMW∞
n .
We prove (1) now; the proof of (2) is deferred until Section 3.4.
Proof of (1). Note that all the tangles involved in the inductive characterization of Yn from
(3.2) are Legendrian with coefficients in F−, and the normalizing factor 1
cn
also belongs to F−. �
The following is the second equality in the statement of Theorem 1.1 from the introduction.
Theorem 3.9. For any Legendrian knot K ⊂ J1R, the n-colored Kauffman polynomial has
Fn,K ∈ F− and satisfies
Fn,K |a−1=0 = R1
n,K .
Proof. Since Yn ∈ BMW−
n we get that Fn,K = FS(K,Yn) ∈ F−, and since ϕ
(
1
cn
Ln
)
= [Yn] the
commutativity of the diagram in Proposition 3.5 together with Proposition 2.8 shows that
Fn,K |a−1=0 = FS(K,Yn)|a−1=0 = R1
S(K, 1
cn
Ln)
= R1
n,K . �
Corollary 3.10. For each n ≥ 2, ϕ
(
1
cn
Ln
)
is a central idempotent in BMW∞
n .
Proof. This follows from Proposition 3.8 since Yn has this property already in BMWn. (See,
e.g., [1, 17].) �
16 J. Murray and D. Rutherford
3.4 Establishing (2) of Proposition 3.8
We now embark on showing ϕ
(
1
cn
Ln
)
= Yn|a−1=0. Throughout this section we will work in
BMW∞
n , but we will simplify notation by writing Ln for ϕ(Ln). We begin with some preparatory
lemmas that provide formulas for Ln that are closer to the inductive formula for Yn from (3.2).
Lemma 3.11 uses the hypothesis that Ln−1 has the crossing absorbing property. This assumption
is later verified to be true (see Proposition 3.13).
Lemma 3.11. Let n ≥ 2 and assume that, in BMW∞
n , Ln−1 has the crossing absorbing proper-
ty (3.1). Then, in BMW∞
n we have
Ln−1βn = Ln−1
1− z
n∑
k=2
s2−k
k−2∑
j=1
−zs2+j−kDn−j,n +Dn−k+1,n
γn.
Proof. We use Lemma 2.7. For each 2 ≤ k ≤ n, let cr`(Ck,n) (respectively crr(Ck,n)) denote
the set of crossings appearing in the left (resp. right) half of Ck,n, and abbreviate Ck := Ck,n.
Then,
Ln−1βn = Ln−1
1− z
n∑
k=2
∑
X⊂cr(Ck)
(−z)|X|rX(Ck)
γn
= Ln−1
1− z
n∑
k=2
∑
X⊂cr`(Ck)
∑
Y⊂crr(Ck)
(−z)|X|+|Y |rX⊔Y (Ck)
γn
= Ln−1
1− z
n∑
k=2
∑
X⊂cr`(Ck)
(−z)|X|
∑
Y⊂crr(Ck)
(−z)|Y |rX⊔Y (Ck)
γn .
Since Ln−1 absorbs crossings Ln−1rX
⊔
Y (Ck) = Ln−1s
k−2−|Y |rX(Dn−k+1,n); see Fig. 8. Fur-
thermore, because there are
(
k−2
|Y |
)
subsets of crr(Ck) having |Y | crossings, summing over |Y | one
obtains
Ln−1
∑
Y⊂crr(Ck)
(−z)|Y |rX⊔Y (Ck) = Ln−1
k−2∑
j=0
(
k − 2
j
)
(−z)jsk−2−jrX(Dn−k+1,n)
= Ln−1(s− z)k−2rX(Dn−k+1,n) = Ln−1s
2−krX(Dn−k+1,n).
Therefore,
Ln−1βn = Ln−1
1− z
n∑
k=2
s2−k
∑
X⊂cr`(Ck)
(−z)|X|rX(Dn−k+1,n)
γn.
It remains to establish the innermost sum satisfies
Ln−1
∑
X⊂cr`(Ck)
(−z)|X|rX(Dn−k+1,n) = Ln−1
k−2∑
j=1
−zs2+j−kDn−j,n +Dn−k+1,n
. (3.4)
When we perform the resolution by subsets of cr`(Ck) we will not be able to feed all of the
remaining crossings into Ln−1 (in fact, when X is the empty set there are no crossings that we
can push into Ln−1). We remedy this by partitioning the nonempty subsets of cr`(Ck). Label the
crossings in cr`(Ck) by ascending z-coordinate as c1, . . . , ck−2, and define χj = {X ⊂ cr`(Ck) :
Legendrian DGA Representations and the Colored Kauffman Polynomial 17
n
n− k + 1
Ln−1 Ln−1
= sk−2−|Y |
Figure 8. An illustration of the identity Ln−1rX
⊔
Y (Ck) = sk−2−|Y |Ln−1rX(Dn−k+1,n), with the
resolved crossings from X
⊔
Y indicated in dotted ovals. The number of crossings in the right half of
rXtY (Ck,n) is k − 2− |Y |.
j = min{i | ci ∈ X}}, for 1 ≤ j ≤ k− 2. Let X ∈ χj be given. By definition cj ∈ X is the lowest
crossing that is resolved in rX(Dn−k+1,n). Isotopy allows us to push the k − 2 − j − (|X| − 1)
remaining crossings lying above cj into Ln−1. Hence, using i = |X|−1 and that the requirement
that X ∈ χj leaves i choices from the k − 2− j crossings above cj to determine X, we have
Ln−1
∑
X∈χj
(−z)|X|rX(Dn−k+1,n) = Ln−1
k−2−j∑
i=0
(
k − 2− j
i
)
(−z)i+1s(k−2−j)−iDn−j,n
= −zLn−1(−z + s)k−2−jDn−j,n (binomial theorem)
= −zLn−1s
2+j−kDn−j,n.
Summing over all j, and adding the termDn−k+1,n for the case whenX is empty, establishes (3.4)
and completes the proof. �
Lemma 3.12. For any 1 ≤ i ≤ n, in BMW∞
n we have Di,nγn =
n−i−1∑
r=0
srDi,n−r.
Proof. This is just applying type II Reidemeister moves to see that
Di,ns
rσn−1σn−2 · · ·σn−r =
{
srDi,n−r, r < n− i,
0, r ≥ n− i,
since the fishtail relation (2.1) may be applied when r ≥ n− i. See Fig. 9. �
=
Figure 9. An Illustration of the Type II Reidemeister moves used in Lemma 3.12.
Proposition 3.8 (2) follows from the following.
Proposition 3.13. For all n ≥ 1, Ln has the following properties in BMW∞
n :
(i) Ln = Yn|a−1=0,
(ii) Ln has the crossing absorbing property (3.1).
18 J. Murray and D. Rutherford
Proof. The proof is by induction on n. In the base case n = 1, (i) is immediate from definitions
and (ii) is vacuous. Assume the result for n − 1. Note that it suffices to establish (i) since the
fact that Yn has the crossing absorbing property and |a−1=0 is a Z(s)-algebra homomorphism
would then allow us to verify (ii) via
Lnσi = (Yn|a−1=0)σi = (Ynσi)|a−1=0 = (sYn)|a−1=0 = sLn.
Showing Ln = Yn|a−1=0 is the following a computation (with Lemmas 3.11 and 3.12 used at the
2nd and 3rd equality):
Ln = Ln−1βn = Ln−1
1− z
n∑
k=2
s2−k
k−2∑
j=1
−zs2+j−kDn−j,n +Dn−k+1,n
γn
= Ln−1
γn − z n∑
k=2
s2−k
k−2∑
j=1
−zs2+j−k
j−1∑
r=0
srDn−j,n−r +
k−2∑
r=0
srDn−k+1,n−r
= Ln−1
γn − z n∑
k=2
s2−k
k−3∑
r=0
k−2∑
j=r+1
−zs2+j−k+rDn−j,n−r +
k−2∑
r=0
srDn−k+1,n−r
= Ln−1
γn − z
n∑
k=2
k−3∑
r=0
k−2∑
j=r+1
−zs4+j−2k+rDn−j,n−r +
n∑
k=2
k−2∑
r=0
s2−k+rDn−k+1,n−r
= Ln−1
γn − z
n−3∑
r=0
n∑
k=r+3
k−2∑
j=r+1
−zs4+j−2k+rDn−j,n−r
+
n−2∑
r=0
n∑
k=r+2
s2−k+rDn−k+1,n−r
])
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
n∑
k=r+3
k−2∑
j=r+1
−zs4+j−2kDn−j,n−r
+
n−1∑
j=r+1
s2−j−1Dn−j,n−r
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
n−2∑
j=r+1
sjDn−j,n−r
n∑
k=j+2
−zs4−2k
+
n−1∑
j=r+1
s2−j−1Dn−j,n−r
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
n−2∑
j=r+1
sjDn−j,n−r
(
z
s4−2n − s2−2j
s2 − 1
)
+
n−1∑
j=r+1
s2−j−1Dn−j,n−r
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
n−2∑
j=r+1
(
sj−2n+3 − s1−j)Dn−j,n−r
Legendrian DGA Representations and the Colored Kauffman Polynomial 19
+
n−1∑
j=r+1
s2−j−1Dn−j,n−r
= Ln−1
(
γn − zD1,2 − z
n−3∑
r=0
sr
[
s2−nD1,n−r
+
n−2∑
j=r+1
(
sj−2n+3 − s1−j + s2−j−1
)
Dn−j,n−r
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
s2−nD1,n−r +
n−2∑
j=r+1
(
sj−2n+3
)
Dn−j,n−r
= Ln−1
γn − zD1,2 − z
n−3∑
r=0
sr
n−1∑
j=r+1
sj−2n+3Dn−j,n−r
= Ln−1
γn − z
s2n−3
s2n−3D1,2 +
n−3∑
r=0
sr
n−1∑
j=r+1
sjDn−j,n−r
= Ln−1
γn − z
s2n−3
s2n−3D1,2 +
n−1∑
i=2
sn−i−1
n−1∑
j=n−i
sjDn−j,i+1
= Ln−1
γn − z
s2n−3
s2n−3D1,2 +
n−1∑
i=2
sn−i−1
i−1∑
j=0
sn−i+jDi−j,i+1
= Ln−1
γn − z
s2n−3
s2n−3D1,2 +
n−1∑
i=2
s2n−2i−1
i−1∑
j=0
sjDi−j,i+1
= Ln−1
γn − z
s2n−3
n−1∑
i=1
s2n−2i−1
i−1∑
j=0
sjDi−j,i+1
= Yn−1
∣∣
a−1=0
·
γn +
z
1− s2n−3a
n−1∑
i=1
as2n−2i−1
i−1∑
j=0
sjDi−j,i+1
∣∣∣∣
a−1=0
= Yn
∣∣
a−1=0
.
At the last equality, we used (3.2). �
4 The n-colored ruling polynomial and representation numbers
In this section, we show that the 1-graded n-colored ruling polynomial agrees with the 1-graded,
total n-dimensional representation number defined in [20]; see Theorem 4.2. After a brief review
of Legendrian contact homology and relevant material from [20], the remainder of the section
contains the proof of Theorem 4.2.
4.1 Review of the Legendrian contact homology DGA
We assume familiarity with the Legendrian contact homology differential graded algebra, (abbrv.
LCH DGA), aka. the Chekanov–Eliashberg algebra, in the setting of Legendrian links in J1M
with M = R or S1, and refer the reader to any of [6, 10, 11, 22, 24] for this background material.
We continue to use coordinates (x, y, z) ∈ J1M = T ∗M ×R = M ×R2, and to view projections
20 J. Murray and D. Rutherford
a+
b1b2
b3
−−
−
Figure 10. A holomorphic disk contributing the term ∂a = ±b1b2t−1b3 + · · · to the differential of A(K).
to S1×R in [0, 1]×R with the left and right boundary identified. The Reeb vector field is ∂
∂z , so
Reeb chords of K are in bijection with double points of the Lagrangian projection aka. the xy-
diagram of K which is the projection to T ∗M . Representation numbers are defined in [20] using
the fully non-commutative version of the LCH DGA associated to a Legendrian knot or link K
equipped with a collection of base points, ∗1, . . . , ∗`, with the requirement that every component
of K has at least one base point. The resulting DGA, notated (A(K), ∂) or (A(K, ∗1, . . . , ∗`), ∂)
when the choice of base points should be emphasized, is an associative, non-commutative algebra
with identity generated over Z by
(i) the Reeb chords of K, denoted b1, . . . , br, and
(ii) invertible generators t±1
1 , . . . , t±1
` corresponding to the base points ∗1, . . . , ∗`.
There are no relations other than tit
−1
i = t−1
i ti = 1. The differential ∂ vanishes on the ti;
for a Reeb chord, a, the differential ∂a is defined via a signed count of rigid holomorphic disks
in T ∗M with boundary on the Lagrangian projection of K and having a single positive boundary
puncture at a and an arbitrary number of negative boundary punctures. Each such a disk u
contributes a term ±w(u) to ∂a where w(u) is the product of base point generators and negative
punctures as they appear in counter-clockwise order along the boundary of the domain of u,
starting from the positive puncture at a. Occurrences of ti appear with exponent ±1 according
to the oriented intersection number of ∂u with ∗i. See Fig. 10. For associating ±1 signs to disks,
we use the conventions as in [19, 20]. Most results of this section concern DGA representations
defined over a field of characteristic 2, and in this case it suffices to work with the version
of A(K) defined over Z/2 where the ±1 signs become irrelevant.
4.2 1-graded representation numbers
In [20], Legendrian invariant m-graded representation numbers are defined for any non-negative
integer m ≥ 0 by considering DGA homomorphisms that are only required to preserve grading
mod m. In the current article, we are concerned only with 1-graded representations, which
we will refer to as ungraded representations since the grading condition becomes vacuous when
m = 1. We review definitions from [20] in the ungraded setting.
Let V be a vector space over a field, F, with char(F) = 2, and let d : V → V be an ungraded
differential on V , i.e., d is just a linear map satisfying d2 = 0. Then, d induces a differential on
the endomorphism algebra
δ : End(V )→ End(V ), δ(T ) = d ◦ T + T ◦ d
making (End(V ), δ) into an ungraded DGA, i.e., δ satisfies δ2 = 0 and δ(T1T2) = δ(T1)T2 +
T1δ(T2). An ungraded representation of a DGA, (A, ∂), on (V, d) is an ungraded DGA homo-
Legendrian DGA Representations and the Colored Kauffman Polynomial 21
morphism
f : (A, ∂)→ (End(V ), δ),
i.e., a ring homomorphism satisfying f(1) = 1 and f ◦∂ = δ ◦f . In the 1-dimensional case where
V = F and d = 0, an ungraded representation on (F, 0) is also called an ungraded augmentation.
For A = A(K), we use the notation Rep1(K, (V, d)) for the set of all ungraded representations
of (A, ∂) on (V, d) and Aug1(K,F) for the set of augmentations to F. In the case where K
is connected with base points ∗1, . . . , ∗` appearing in order starting at ∗1 and following the
orientation of K, given a subset T ⊂ GL(V ) we will use the notation Rep1(K, (V, d), T ) to denote
the set of those f ∈ Rep1(K, (V, d)) such that f(t1 · · · t`) ∈ T . In particular, Rep1(K, (V, d)) =
Rep1(K, (V, d),GL(V )).
Definition 4.1. Let Fq denote the finite field of order q with q a power of 2. The (1-graded)
total n-dimensional representation number of K is defined by
Rep1
(
K,Fnq
)
:=
∣∣End
(
Fnq
)∣∣−rb(K)/2|GL(n,Fq)|−`
∣∣Rep1
(
K,
(
Fnq , 0
))∣∣
=
(
qn
2)−rb(K)/2
(
qn(n−1)/2
n∏
m=1
(
qm − 1
))−` ∣∣Rep1
(
K,
(
Fnq , 0
))∣∣,
where rb(K) is the number of Reeb chords of K and ` is the number of basepoints.
That Rep1
(
K,Fnq
)
only depends on the Legendrian isotopy type of K is established in [20,
Proposition 3.10]. The following theorem shows that the 1-graded n-colored ruling polynomial
and the total n-dimensional representation numbers of K are equivalent Legendrian invariants.
Theorem 4.2. Let K ⊂ J1R be a (connected) Legendrian knot and Fq a finite field of order q
with characteristic 2. Then,
R1
n,K(q) = Rep1(K,Fq),
where R1
n,K(q) = R1
n,K(s)
∣∣
s=q1/2
.
The proof of Theorem 4.2 rests on the following relation between representation counts and
reduced ruling polynomials which we will establish in Sections 4.3–4.5 using extensions of results
from [19] and [20]. In [20, Section 4], a “path subset”3 Bβ ⊂ GL(n,Fq) is associated to a reduced4
positive permutation braid, β ∈ Sn.
Lemma 4.3. Let β ∈ Sn be an n-stranded reduced positive permutation braid, and suppose that
the (connected) Legendrian knot K ⊂ J1R has its front diagram in plat position5 and is equipped
with ` base points where ` is the number of right cusps of K. Then,∣∣Rep1
(
K,
(
Fnq , 0
)
, Bβ
)∣∣ = |GL(n,Fq)|`−1qn(n−1)/2(q − 1)nqn
2rb(K)/2qλ(β)/2R̃S(K,β)(z),
where rb(K) is the number of Reeb chords of K, λ(β) is the length of β, i.e., the number of
crossings of β, and z = q1/2 − q−1/2.
3The path subset Bβ is the subset of GL(n,Fq) arising from specializing the path matrix P xyβ using arbitrary
ring homomorphisms from A(β) to Fq. The path matrix P xyβ is a matrix whose entries belong to A(β) and
record certain left-to-right paths through the xy-diagram of β that reflect the possible behavior of boundaries of
holomorphic disks bordering β from above. See [20, Section 4.1]. Note that [20] uses the opposite convention for
composing Legendrian n-tangles, with α · β defined as α stacked to the left of β. Consequently, for a permutation
β ∈ Sn, the permutation braid β ⊂ J1S1 used in the present article corresponds to β−1 ⊂ J1S1 in [20]. As
a result, the notations Bβ and P xyβ used here correspond to Bβ−1 and P xy
β−1 in [20].
4A positive permutation braid β ⊂ J1S1 is reduced if its front diagram corresponds to a reduced braid word,
i.e., one where the product σiσi+1σi does not appear for any i.
5A front diagram is in plat position if all left cusps appear at a common x-coordinate at the far left of the
diagram and all right cusps appear at a common x-coordinate at the far right of the diagram.
22 J. Murray and D. Rutherford
Proof of Theorem 4.2. Since both sides of the equation are Legendrian isotopy invariant,
we may assume K is in plat position. In [20, Proposition 4.14], it is shown that GL(n,Fq) =
tβ∈SnBβ. (Actually, this coincides with the Bruhat decomposition of GL(n,Fq).) Thus,
Rep1
(
K,
(
Fnq , 0
)
,GL(n,Fq)
)
= tβ∈SnRep1
(
K,
(
Fnq , 0
)
, Bβ
)
,
and using Lemma 4.3 and Definition 2.5 we compute
Rep1
(
K,Fnq
)
=
(
qn
2)−rb(K)/2|GL(n)|−`
∑
β∈Sn
Rep1
(
K,
(
Fnq , 0
)
, Bβ
)
= |GL(n)|−1qn(n−1)/2(q − 1)n
∑
β∈Sn
qλ(β)/2R̃S(K,β)(z)
=
(
qn(n−1)/2
n∏
m=1
(
qm − 1
))−1
qn(n−1)/2(q − 1)n
∑
β∈Sn
qλ(β)/2R̃S(K,β)(z)
=
(
n∏
m=1
(
qm − 1
q − 1
))−1 ∑
β∈Sn
qλ(β)/2R̃S(K,β)(z)
=
1
cn
∑
β∈Sn
qλ(β)/2R̃S(K,β)(z) = R1
n,K(q).
(The notation cn is as in Definition 2.5 with s = q1/2.) �
4.3 Strategy of the proof of Lemma 4.3
The relation between counts of representations on
(
Fnq , 0
)
and reduced ruling polynomials stated
in Lemma 4.3 is based on refinements of two results:
1. In [20, Theorem 6.1], it is shown that for any n-stranded reduced positive permutation
braid β ∈ Sn, (using 1 base point on K and a particular xy-diagram of S(K,β)) when
char(F) = 2 there is a bijection⊔
d
Rep1
(
K,
(
Fn, d
)
, Bβ
)
←→ Aug1(S(K,β),F), (4.1)
where the union is over all strictly upper triangular differentials on Fn.
2. In [19, Theorem 3.2], for any Legendrian K ′ ⊂ J1R (using a particular xy-diagram of K ′)
the set of augmentations of K ′ is decomposed into pieces
Aug1(K ′,F) =
⊔
ρ
(F∗)a(ρ) × Fb(ρ), (4.2)
where the disjoint union is indexed by all normal rulings of K ′ and the exponents a(ρ)
and b(ρ) are specified by the combinatorics of ρ. Theorem 1.1 of [19] then applies the
decomposition to relate the Legendrian invariant augmentation numbers with the ruling
polynomial.
The idea behind the proof of Lemma 4.3 is then to check that the subset of Aug1(S(K,β),F)
corresponding under (4.1) to Rep1
(
K,
(
Fn, 0
)
, Bβ
)
, i.e., those representations with d = 0, is the
part of the disjoint union (4.2) with K ′ = S(K,β) that is indexed by reduced normal rulings
of S(K,β). This is roughly what we shall do. However, complications arise as different (but
Legendrian isotopic) xy-diagrams for the satellite S(K,β) having different (but stable tame
Legendrian DGA Representations and the Colored Kauffman Polynomial 23
isomorphic) DGAs are used in (4.1) and (4.2). As a result, we need to also keep track of
the way the set Aug1(S(K,β),F) changes when transitioning between these different diagrams
for S(K,β). This contributes to the factor appearing in front of R̃S(K,β) in the statement of
Lemma 4.3.
Remark 4.4. In the case of m-graded representations with m 6= 1, d = 0 is the only term in
the disjoint union (4.1) for grading reasons.
4.4 Four diagrams for the satellite, S(K,β)
In establishing Lemma 4.3 we will make use of four different (but Legendrian isotopic) versions
of the satellite S(K,β). The four xy-diagrams are denoted S1
xy(K,β), S2
xy(K,β), S1
xz(K,β),
S2
xz(K,β) and will be defined momentarily; see Fig. 12. As a preliminary, we consider xz-diag-
rams (front projection) and xy-diagrams (Lagrangian projection) for the companion K ⊂ J1R
and pattern β ⊂ J1S1.
4.4.1 Diagrams for K and β
For the companion knot K ⊂ J1R, apply Ng’s resolution procedure (see [22]) so that the xy-
diagram for K is related to the front projection of K by placing the strand with smaller slope
on top at crossings and adding an extra loop at right cusps. Enumerate the Reeb chords of K
by a1, . . . , am, and c1, . . . , c` where the ai correspond to crossings of the front projection of K
and the ci are the extra crossings near right cusps that arise from the resolution procedure.
Choose an initial base point ∗ of K not located on any of the loops at right cusps and at a point
where the front diagram of K is oriented left-to-right. For convenience, we assume the c1, . . . , c`
are enumerated in the order they appear when following along K according to its orientation,
starting at ∗.
For the braid β ⊂ J1S1, we form an xy-diagram as indicated in Fig. 11 having ` dips in
addition to the original crossings coming from the front projection of β. This is done by applying
the resolution procedure to the front diagram of β as in [27, Section 2.2] (this amounts to adding
a dip to the right of the crossings of β), and then adding ` − 1 extra dips to the xy-diagram.
In addition, a collection of n basepoints, ∗1, . . . , ∗n, are placed immediately to the left of the
crossings of β, one on each strand. Note that the addition of the dips can be accomplished by
Legendrian isotopy as indicated in [31, Section 3.1], and we have used a minor variation on the
resolution procedure of [27, Section 2.2] so that the dip arising from the resolution procedure
has the same form as the others. Numbering the dips 1, . . . , ` as they appear from left to right
in T ∗S1 (viewed as [0, 1] × R with boundaries identified) the k-th dip consists of two groups
of crossings xki,j and yki,j for 1 ≤ i < j ≤ n that appear in the left and right half of the dip
respectively. For 1 ≤ k ≤ `, we collect these crossings as the entries of strictly upper triangular
matrices denoted Xk and Yk. The crossings that correspond to xz-crossings of β are labeled
p1, . . . , pλ, and DGA generators associated to the basepoints ∗1, . . . , ∗n will be labelled s1, . . . , sn.
In constructing the diagrams S1
xy(K,β) and S2
xy(K,β) and comparing their DGAs, it is
useful to be able to move the locations of dips around via a Legendrian isotopy. This may
be accomplished by converting an isotopy φt : S
1 → S1, 0 ≤ t ≤ 1, to an ambient contact
isotopy
Φt : J
1S1 → J1S1, Φt(x, y, z) =
(
φt(x),
y
φ′t(x)
, z
)
, 0 ≤ t ≤ 1. (4.3)
In particular, given any cyclically ordered collection of points x1, . . . , x` ∈ S1, after a Legen-
drian isotopy the Xk and Yk crossings can be arranged to appear above an arbitrarily small
neighborhood of xk in S1 for 1 ≤ k ≤ `.
24 J. Murray and D. Rutherford
z
x
y
x
X1 Y1 X2 Y2
;
∗3
∗2
∗1
Figure 11. A Lagrangian diagram for the positive braid β = σ2σ1σ2 with ` = 2 dips.
4.4.2 Diagrams for Legendrian satellites
The Legendrian satellite S(K,β) ⊂ J1R is formed by scaling the y and z coordinates of J1S1
so that β sits in a small neighborhood, N0 ⊂ J1S1, of the 0-section and then applying a contac-
tomorphism Ψ: N0 → N(K) of N0 onto a Weinstein tubular neighborhood of K. Two general
methods for forming xy-diagrams of satellites are as follows:
• The xy-method. Use the orientations of K and R2 to identify an (immersed) annular
neighborhood, Nxy, of the xy-diagram of K with S1 × (−ε, ε). Then, (after scaling the
y-coordinate appropriately) place an xy-diagram for β into Nxy. As Ψ can be chosen to
preserve the Reeb vector field (see [16]), this indeed produces an xy-diagram for S(K,β).
• The xz-method. First form an xz-diagram for S(K,β) as in Section 2.3: start with the
n-copy of K (this is n-parallel copies of K shifted a small amount in the z-direction), then
insert the xz-projection of β at the location of the initial base point of K. Finally, produce
an xy-diagram for S(K,β) by applying Ng’s resolution procedure.
Definition of S1
xy(K,β): To form S1
xy(K,β) expand ∗ to a cluster of base points ∗1, . . . , ∗`
all appearing in order (according to the orientation of K) in a small neighborhood of ∗. Then,
form the satellite with the xy-diagram of β described above in such a way that the crossings
p1, . . . , pλ from β and the crossings from X1, Y1 appear in a neighborhood of ∗1, and the crossings
from Xk, Yk appear in a neighborhood of ∗k for 2 ≤ k ≤ `.
Definition of S2
xy(K,β): This xy-diagram is formed from S1
xy(K,β) by using a Legendrian
isotopy of β (constructed from an isotopy of S1 as in (4.3)) to relocate each collection of Xi
crossings to a neighborhood of a right cusp of K (so that exactly one Xi appears near each
right cusp); relocate each group of Yi crossings to a neighborhood of a left cusp; and leave the
crossings p1, . . . , pλ of β (and the basepoints ∗1, . . . , ∗n) in place at the location of the initial
base point ∗ for K. (Recall that ` is both the number of dips and the number of right cusps
of K.)
Definition of S1
xz(K,β) and S2
xz(K,β): Both S1
xz(K,β) and S2
xz(K,β) are formed us-
ing the xz-method. The only difference between the two is the placement of base points.
For S1
xz(K,β), base points ∗1, . . . , ∗n appear, one on each parallel strand of the n-copy, just
before β (with respect to the orientation of K). For S2
xz(K,β), we have base points ∗1, . . . , ∗c,
one on each component of S(K,β) placed on a loop near some right cusp of the component.
See Fig. 12.
4.4.3 DGA generators
We set notations for the generators of the DGAs arising from the various xy-diagrams of S(K,β)
that have been defined. Many generators are indexed with a pair of subscripts i, j. Such
a subscript indicates that at the overstrand of the crossing belongs to the i-th copy of K and the
understrand belongs to the j-th copy of K. Here, outside of an arc A ⊂ K where the p1, . . . , pλ
Legendrian DGA Representations and the Colored Kauffman Polynomial 25
S1
xy(K,β) S2
xy(K,β)
S1
xz(K,β) S2
xz(K,β)
Figure 12. The Lagrangian (xy)-diagrams S1
xy(K,β), S2
xy(K,β), S1
xz(K,β), and S2
xz(K,β) where K is
a Legendrian trefoil and β = σ1 ∈ S2.
crossings of β appear, S(K,β) consists of n copies of K \A, which we label from 1 to n according
to the descending order of their y-coordinates at the boundary of A.
Generators of A(S1
xy(K,β)) and A(S2
xy(K,β)): The generating sets for these DGAs
are in bijection. Both contain the DGA of β as a sub-DGA, and this accounts for generators of
the form p1, . . . , pλ, xki,j , y
k
i,j for 1 ≤ k ≤ ` and 1 ≤ i < j ≤ n as well as invertible generators
s1, . . . , sn associated to the base points ∗1, . . . , ∗n on β. In addition, for each of the Reeb chords
a1, . . . , am, and c1, . . . , c` of K there are n2 Reeb chords for S1
xy(K,β) and S2
xy(K,β) that we
denote by ak1i,j and ck2i,j , 1 ≤ k1 ≤ m, 1 ≤ k2 ≤ `, 1 ≤ i, j ≤ n.
Generators of A(S1
xz(K,β)) and A(S2
xz(K,β)): Both of these diagram have Reeb
chords
• p1, . . . , pλ from the xz-crossings of β;
• ak1i,j , 1 ≤ k1 ≤ m, 1 ≤ i, j ≤ n, from the xz-crossings of K;
• yk2i,j , 1 ≤ k2 ≤ `, 1 ≤ i < j ≤ n, from crossings near left cusps;
• ck2i,j , 1 ≤ k2 ≤ `, 1 ≤ j ≤ i ≤ n, from crossings near right cusps.
The invertible generators coming from base points will be denoted as s1, . . . , sn for S1
xz(K,β)
and as r1, . . . , rc for S2
xz(K,β).
In particular, note that for any of the four diagrams for S(K,β) there is a collection of
generators of the form yki,j which we will refer to as Y -generators.
Definition 4.5. Let K ′ denote one of S1
xy(K,β), S2
xy(K,β), S1
xz(K,β), or S2
xz(K,β). We denote
by
Aug1(K ′,F)Y=0
the set of augmentations of A(K ′) to F that map all Y -generators to 0.
26 J. Murray and D. Rutherford
The subsets Aug1(K ′,F)Y=0 ⊂ Aug1(K ′,F) play an important role in the proof Lemma 4.3
as they correspond to representations with d = 0 as well as to reduced normal rulings.
Remark 4.6. Computations of the differential for similar DGAs of 1-dimensional Legendrian
satellites are presented in detail in several places, and it is not difficult to extend these compu-
tations to give complete formulas for differentials in the present setting. See especially [20] for
satellites formed via the xy-method and [24] and [25] for satellites formed via the xz-method.
Rather than giving a complete description of differentials here, we state several partial formulas
as they become useful in the following proofs.
4.5 Representations and augmentations of the satellite
We will use the following mild variation of [20, Theorem 6.1] to transition between n-dimensional
representations and augmentations of satellites. As in the construction of S1
xy(K,β), expand the
initial base point ∗ of K into a cluster of base points ∗1, . . . , ∗` and consider the DGA (A(K), ∂)
with invertible generators t1, . . . , t`.
Proposition 4.7. Let β be a reduced positive permutation braid, and construct S1
xy(K,β) as
above. There is a bijection{
f ∈ Rep1
(
K,
(
Fn, 0
))
| f(t1) ∈ Bβ and f(ti) ∈ N+ for i ≥ 2
}
↔ Aug1(S1
xy(K,β),F)Y=0,
where Bβ is the path subset of β and N+ ⊂ GL(n,F) is the group of upper triangular matrices
with 1’s on the diagonal.
Proof. The proof is similar to [20, Theorem 6.1] which implies the case of only 1 base point.
We sketch the argument and highlight the modifications to the proof for the case of more than
one basepoint. To avoid considering signs, we only treat the case where char(F) = 2 (which
is the only case needed for Lemma 4.3). This allows us to work with the LCH DGA defined
over Z/2 rather than over Z (since any representation with char(F) = 2 factors through the
change of coefficients map from the DGA over Z to the DGA over Z/2).
To relate the differential on D : A
(
S1
xy(K,β)
)
→ A
(
S1
xy(K,β)
)
to the differential ∂ on A(K)
consider a Z/2-algebra homomorphism
Φ: A(K)→ Mat
(
n,A
(
S1
xy(K,β)
))
that sends Reeb chords ak or ck to the corresponding n×n matrices of Reeb chords Ak =
(
aki,j
)
or Ck =
(
cki,j
)
and satisfies
Φ(t1) = P xyβ , Φ(tk) = (I +Xk), for 2 ≤ k ≤ `,
where P xyβ and P xzβ denote the xy- and xz-path matrices of β as defined in [20, Section 4.1].
Over Z/2, with the differential D on A
(
S1
xy(K,β)
)
extended entry-by-entry to
Mat
(
n,A
(
S1
xy(K,β)
))
we have the identities,
DYk = Y 2
k , 1 ≤ k ≤ `, and (4.4)
D ◦ Φ(x) = Φ ◦ ∂(x) +O(Y ), for any generator x ∈ A(K), (4.5)
where O(Y ) denotes a term belonging to the 2-sided ideal generated by the Y -generators. This
is essentially as in [20, Section 5]; see especially Proposition 5.2 and Corollary 5.3 of [20]. For
generalizing to the case of more than one base point, note that, for 1 ≤ k ≤ `, Φ(tk) is the left-
to-right xy-path matrix for the part of the xy-diagram of β that sits in a neighborhood of the
basepoint ∗k on K. As a result, the entries of Φ(tk) (resp. Φ
(
t−1
k
)
) record the possibly negative
Legendrian DGA Representations and the Colored Kauffman Polynomial 27
punctures that boundaries of “thick disks” of S1
xy(K,β) can have when they pass through the
location of ∗k on K in a way that agrees (resp. disagrees) with the orientation of K; see [20,
Sections 4.1 and 5.2].
Now, the bijection from the statement of the proposition arises from associating to an aug-
mentation ε ∈ Aug1
(
S1
xy(K,β),F
)
Y=0
the matrix representation f : A(K) → Mat(n,F) given
by f = ε ◦ Φ. From here (4.4) and (4.5) can then be used to show that under the assumption
that ε(Y ) = 0, the augmentation equation ε ◦D = 0 is equivalent to the representation equation
f ◦ ∂ = 0, cf. [20, Theorem 6.1]. (Note that the hypothesis that β is a reduced positive permu-
tation braid is used as in [20] to see that the equation ε(DΦ(t1)) = 0 is equivalent to having
ε ◦D(x) = 0 for all generators of the form pi or x1
i,j .) �
We are now prepared to give the proof of Lemma 4.3.
Proof of Lemma 4.3.
Step 1. Establish that
∣∣Rep1
(
K,
(
Fnq , 0
)
, Bβ
)∣∣ =
|GL(n)|`−1
(qn(n−1)/2)`−1
·
∣∣Aug1
(
S1
xy(K,β),Fq
)
Y=0
∣∣.
Keeping in mind that Rep1(K, (Fnq , 0), Bβ) is defined in terms of the DGA A(K, ∗1, . . . , ∗`)
of K equipped with ` base points ∗1, . . . , ∗`, let Rep1
(
A(K, ∗),
(
Fnq , 0
)
, Bβ
)
denote the corre-
sponding set of representations with respect to the DGA of K equipped with only the one initial
base point ∗, i.e., the set of ungraded representations f : (A(K, ∗), ∂) → (Mat(n,Fq), 0) having
f(t) ∈ Bβ. The differential of a Reeb chord b of K in A(K, ∗1, . . . , ∗`) is obtained from its
differential in A(K, ∗) by replacing all occurrences of t by the product t1 · · · t`. Consequently,∣∣Rep1
(
K,
(
Fnq , 0
)
, Bβ
)∣∣ = |GL(n)|`−1 ·
∣∣Rep1
(
A(K, ∗),
(
Fnq , 0
)
, Bβ
)∣∣,
where the factor |GL(n)|`−1 arises as the number of ways to factor a given matrix f(t) ∈ Bβ
into a product f(t1) · · · f(t`) with f(ti) ∈ GL(n) for 1 ≤ i ≤ `. On the other hand, using
Proposition 4.7 gives∣∣Aug1(S1
xy(K,β),Fq)Y=0
∣∣ =
(
qn(n−1)/2
)`−1∣∣Rep1
(
A(K, ∗),
(
Fnq , 0
)
, Bβ
)∣∣,
where in this case the factor
(
qn(n−1)/2
)`−1
arises as the number of ways to factor a given
matrix f(t) ∈ Bβ into a product f(t1) · · · f(t`) with f(t1) ∈ Bβ and f(ti) ∈ N+ for 2 ≤
i ≤ `. Here, we use the connection with the Bruhat decomposition from [20, Section 4.3] so
that Bβ has the form BSβB where Sβ is a permutation matrix and B is the group of invertible
upper triangular matrices. Therefore, given f(t) ∈ Bβ and strictly upper-triangular matrices
f(t2), . . . , f(t`) ∈ N+ there is a unique element f(t1) ∈ Bβ so that f(t) = f(t1)f(t2) · · · f(t`).
Step 2. Establish that∣∣Aug1
(
S1
xy(K,β),Fq
)
Y=0
∣∣ =
∣∣Aug1
(
S2
xy(K,β),Fq
)
Y=0
∣∣.
A Legendrian isotopy of β as in (4.3) can be used to produce a Legendrian isotopy of S(K,β)
that moves the Xk and Yk crossings around the annular neighborhood Nxy of the Lagrangian
projections of K. In particular, this procedure leads to a Legendrian isotopy Λt, 0 ≤ t ≤ 1,
from S1
xy(K,β) to S2
xy(K,β). Note that the crossings p1, . . . , pλ and the base points ∗1, . . . , ∗n
can be assumed to remain in place during the isotopy. Moreover, since the xy-diagram of β has
no vertical tangencies, by scaling the y- and z- coordinates by an appropriately small factor,
it can be assumed that, for all 0 ≤ t ≤ 1, the xy-diagram of Λt does not have self-tangencies.
As a result, the Reeb chords of Λt appear in continuous 1-parameter families parametrized by
28 J. Murray and D. Rutherford
t ∈ [0, 1], and (identifying the corresponding generators of all A(Λt)) the differential remains
constant except for a finite number of handleslide disk bifurcations. These occur when an Xk
or Yk crossing of β passes over or under another strand of the satellite as in Move I of [11,
Section 6]. During the move, there are three Reeb chords, one that is a crossing of β and
two more of the form aki,j or cki,j , that all come together at a triple point. Label the three
Reeb chords as x, y, z so that their lengths (i.e., the difference of z-coordinates at endpoints)
satisfy h(x) > h(y) > h(z), and note that z is the crossing from β. The DGAs before and
after the triple point move are related by a DGA isomorphism φ : (A, ∂)→ (A′, ∂′) that maps x
to an element of the form x ± yz or x ± zy and fixes all other generators. In particular,
φ restricts to the identity on the sub-algebra generated by the Y -generators. Composing all
of the DGA isomorphisms from handleslide disks, we see that there is a DGA isomorphism
ϕ :
(
A
(
S1
xy(K,β)
)
, ∂1
)
→
(
A
(
S2
xy(K,β)
)
, ∂2
)
that restricts to the identity on all Y -generators.
As a result, ϕ∗ : Aug1
(
S2
xy(K,β),Fq
)
→ Aug1
(
S1
xy(K,β),Fq
)
, ϕ∗ε = ε ◦ ϕ induces the required
bijection between Aug1
(
S1
xy(K,β),Fq
)
Y=0
and Aug1
(
S2
xy(K,β),Fq
)
Y=0
.
Step 3. Establish that∣∣Aug1
(
S2
xy(K,β),Fq
)
Y=0
∣∣ =
(
qn(n−1)/2
)`∣∣Aug1
(
S1
xz(K,β),Fq
)
Y=0
∣∣. (4.6)
The generators of S1
xz(K,β) are identified with a subset of the generators of S2
xy(K,β), and
this leads to an algebra inclusion i : A
(
S1
xz(K,β)
)
→ A
(
S2
xy(K,β)
)
. In fact, it is not hard to
check that i is a DGA homomorphism; see [25, Proposition 4.23] for a detailed explanation in the
case where β is the identity braid. The difference between the two DGAs is that A
(
S2
xy(K,β)
)
has additional generators of the form cki,j and xki,j with 1 ≤ k ≤ `, 1 ≤ i < j ≤ n that
A
(
S1
xz(K,β)
)
does not have. Equation (4.6) then follows from:
Claim: Given any ε ∈ Aug1
(
S1
xz(K,β),Fq
)
Y=0
and arbitrary values ε′
(
cki,j
)
∈ Fq, there exists
a unique augmentation ε′ ∈ Aug1
(
S2
xy(K,β),Fq
)
Y=0
extending these values and restricting to ε
on A
(
S1
xz(K,β)
)
.
Given ε and ε′
(
cki,j
)
we need to show that there are unique values ε′
(
xki,j
)
for which the
equations
ε′ ◦ ∂
(
cki,j
)
= 0 and ε′ ◦ ∂
(
xki,j
)
= 0, 1 ≤ k ≤ `, 1 ≤ i < j ≤ n
hold. Since ∂xki,j belongs to the 2-sided ideal generated by the Y -generators, and ε vanishes
on all Y -generators, the equations ε′ ◦ ∂
(
xki,j
)
= 0 are satisfied. Note that (again computing
over Z/2)
∂Ck = (I +Xk)
±1 +Wk,
where Wk denotes a matrix with entries in the subalgebra generated by A1
xz(S(K,β)) and by
the cki,j generators. The map Fn(n−1)/2
q → Fn(n−1)/2
q that sends a collection of values
(
ε′
(
xki,j
))
i<j
to the above diagonal part of ε′
(
(I + Xk)
±1
)
is a bijection. Hence, there is a unique way to
choose ε′
(
xki,j
)
so that the equations ε ◦ ∂cki,j = 0, 1 ≤ i < j 6= n hold (since this is equivalent to
having ε′
(
(I +Xk)
±1
)
= ε′(Wk) hold on the upper triangular entries.)
Step 4. Establish that∣∣Aug1
(
S1
xz(K,β),Fq
)
Y=0
∣∣ = (q − 1)n−c
∣∣Aug1
(
S2
xz(K,β),Fq
)
Y=0
∣∣. (4.7)
First, note that when the locations of basepoints are moved around the number of Y = 0
augmentations does not change. (This is because when a basepoint si moves through an xy-
crossing a, the DGAs before and afterward are related by an isomorphism φ that maps a to
an element of the form s±1
i a or as±1
i (depending on orientation of K and whether si passes
Legendrian DGA Representations and the Colored Kauffman Polynomial 29
the upper or lower endpoint of a) and fixes all other generators; see [24, Theorem 2.20]. In
particular, the induced bijection between augmentation sets, φ∗, preserves the Y = 0 subsets
(since ε(a) = 0 if and only if ε
(
t±1
i a
)
= 0).
To understand the effect of changing the number of basepoints, suppose that K1 and K2 are
two xy-diagrams, identical except that on K1 a collection of base points with corresponding
generators u1, . . . , us appears near the location of a single base point u of K2. Arguing as
in Step 1, shows that there is a DGA map φ : A(K2) → A(K1) with φ(u) = u1 · · ·us and
φ(x) = x for all other generators, and moreover φ∗ : Aug1(K1,Fq)→ Aug1(K2,Fq) is surjective
and (q − 1)s−1-to-1. (Here, (q − 1)s−1 represents the number of ways to factor an element of F∗q
into a product of s elements in F∗q .) Since φ is the identity on Reeb chords, when applied
to S(K,β), φ∗ restricts to a surjective, (q − 1)s−1-to-1 map between the Y = 0 augmentation
sets. Starting with S1
xz(K,β) and applying this procedure repeatedly with the collection of base
points on each component (keeping in mind that S1
xz(K,β) has n base points while S2
xz(K,β)
has c) leads to the formula (4.7).
Step 5. Establish a decomposition of the form
Aug1
(
S2
xz(K,β),Fq
)
Y=0
=
⊔
ρ
Wρ,
where the disjoint union is over reduced normal rulings, ρ, and
|Wρ| = (q − 1)j(ρ)+c · q
1
2
(−j(ρ)+rb(S(K,β)))
with c the number of components of S(K,β) and rb(S(K,β)) the number of Reeb chords
of S2
xz(K,β).
A decomposition of the entire augmentation variety Aug1
(
S2
xz(K,β),Fq
)
(without imposing
the Y = 0 condition) as
⊔
Wρ with the disjoint union over all normal rulings (without the
reduced condition) is established in Theorem 3.4 of [19]. The statement of that theorem implies
that |Wρ| = (q − 1)j(ρ)−cqb(ρ) where b(ρ) is the number of “returns”6 of ρ plus the number of
right cusps of ρ, and Lemma 5 of [26] shows that b(ρ) = 1
2(−j(ρ)+rb(S(K,β))). Thus, it suffices
to show that an augmentation ε ∈ Aug1
(
S2
xz(K,β),Fq
)
belongs to Wρ with ρ reduced if and only
if the Y = 0 condition is satisfied.
A summary of the construction of the decomposition Aug1
(
S2
xz(K,β),Fq
)
=
⊔
Wρ is as
follows. Let K ′ ⊂ J1R be a Legendrian in plat position whose DGA is computed from resolving
the front projection of K ′ and positioning a single base point on each component of K ′ in the loop
near some chosen right cusp. The DGA of S2
xz(K,β) is of this required type. For such a K ′, the
article [19] considers objects called Morse complex sequences (MCSs) that consist of a sequence of
chain complexes and formal handleslide marks (which are vertical segments on the front diagram
of K ′) subject to several axioms motivated by Morse theory. Section 5 of [19] gives a bijection
between Aug1(K ′,Fq) and the set of “A-form”7 MCSs for K ′, denoted here MCSA(K ′), where
an augmentation ε corresponds to an A-form MCS with one handleslide mark just to the left of
every crossing or right cusp, x, with ε(x) 6= 0 with the handleslide coefficient determined by the
value ε(x). In Section 4.1 of [19], another class of MCSs called “SR-form” MCSs are considered;
we will denote the set of all SR-form MCSs for K ′ as MCSSR(K ′). Each SR-form MCS, C,
has an associated normal ruling ρ of K ′, and all handleslide marks of C appear in collections
of a standard form near switches, returns, and right cusps of ρ. In particular, at every switch
of ρ, C must have a collection of handleslides with non-zero coefficients; returns and right cusps
6Given a normal ruling ρ for a Legendrian link K′, the xz-crossings of K′ that are not switches are either
departures or returns. At a departure (resp. return) the normality condition holds to the left (resp. right) of the
crossing but not to the right (resp. left) of the crossing. See [26, Section 3].
7An MCS is in A-form if its handleslides only appear in specified locations on the front diagram of K′ to the
left of crossings and right cusps. See [19, Section 5].
30 J. Murray and D. Rutherford
may or may not have handleslides. In Section 6 of [19] a bijection Ψ: MCSA(K ′)→ MCSSR(K ′)
and its inverse Φ = Ψ−1 are constructed. By definition, ε ∈ Wρ if the corresponding A-form
MCS, Cε, is such that the SR-form MCS Ψ(Cε) has associated normal ruling ρ.
Note that for an A-form or SR-form MCS every handleslide has some associated xz-crossing
or right cusp. (For an SR-form MCS with normal ruling ρ, associated crossings can only be
switches or returns of ρ and a single crossing may have more than one handleslide associated to
it.) Let MCSA(S(K,β))Y=0 and MCSSR(S(K,β))Y=0 denote those A-form and SR-form MCSs
for S2
xz(K,β) that do not have any handleslide marks associated to xz-crossings corresponding
to the Y -generators. (These are the groups of n(n − 1)/2 crossings that appear in S2
xz(K,β)
near the location of left cusps of K.) Step 5 is then completed by the following.
Lemma 4.8. The above constructions from [19] restrict to bijections
Aug1
(
S2
xz(K,β),Fq
)
Y=0
↔ MCSA(S(K,β))Y=0 ↔ MCSSR(S(K,β))Y=0
and an SR-form MCS C belongs to MCSSR(S(K,β))Y=0 if and only if the associated normal
ruling ρ is reduced.
Proof. The first bijection is clear since an A-form MCS has no handleslides at Y crossings if
and only if the corresponding augmentation vanishes on Y -generators. Turning to the second
bijection, given an A-form or SR-form MCS, C, let i(C) denote the first (from left to right)
xz-crossing of S2
xy(K,β) that has a handleslide associated to it. The defining requirement for
both MCSA(S(K,β))Y=0 and MCSSR(S(K,β))Y=0 is equivalent to i(C) not being one of the Y -
crossings. (The Y -crossings all appear to the left of the other xz-crossings of S2
xz(K,β) since K
is in plat position.) Examining the definition of Ψ and Φ in Section 6 of [19], it is straightforward
to see that i(Ψ(C)) = i(C) and i(Φ(C)) = i(C), so that Ψ and Φ restrict to provide the bijection
MCSA(S(K,β))Y=0 ↔ MCSSR(S(K,β))Y=0.
For the final statement of the lemma, note that a normal ruling of S2
xz(K,β) is reduced if
and only if it has no switches at Y -crossings. See [24, Lemma 3.2]. Moreover, if a Y -crossing is
a return then there must be another Y -crossing somewhere to its left that is a switch. (If there
are no switches in the Y -crossings then, it is easy to see that all Y -crossings are departures.)
Thus, for C in SR-form, i(C) is a Y -crossing if and only if the corresponding ruling is not
reduced. �
Step 6. Completion of the proof.
Combining the identities from Steps 1–5, we have∣∣Rep1
(
K,
(
Fnq , 0
)
, Bβ
)∣∣ = |GL(n)|`−1qn(n−1)/2(q − 1)n−c
∑
ρ
(q − 1)j(ρ)+cq
1
2
(−j(ρ)+rb(S(K,β)))
= |GL(n)|`−1qn(n−1)/2(q − 1)nq
1
2
rb(S(K,β))
∑
ρ
(q1/2 − q−1/2)j(ρ)
= |GL(n)|`−1qn(n−1)/2(q − 1)nqn
2rb(K)/2qλ(β)/2R̃S(K,β)(z),
where the summations are over all reduced normal rulings and at the last equality we used that
the number of Reeb chords of S2
xz(K,β) is n2 · rb(K) + λ(β). �
5 The multi-component case
For simplicity, we have restricted the focus of this article to the case where K is a connected
Legendrian knot. We close with a discussion of an appropriate modification of Theorem 1.1 for
the case of Legendrian links with multiple components.
Legendrian DGA Representations and the Colored Kauffman Polynomial 31
When K = tci=1Ki is a Legendrian link with c components, for ~n = (n1, . . . , nc) with ni ≥ 1
one can consider the ~n-colored Kauffman polynomial defined by satelliting each Ki with the
symmetrizer from the ni-stranded BMW algebra in a multi-linear manner. With the ~n-colored
1-graded ruling polynomial defined by the analogous modification,
R1
~n,K(q) =
(
c∏
i=1
1
cni
) ∑
~β∈Sn1×···×Snc
(
c∏
i=1
qλ(βi)/2
)
R̃
S(K,~β)
(z)|z=q1/2−q−1/2 ,
the second equality of Theorem 1.1, modified to read R1
~n,K(z) = F~n,K(a, q)|a−1=0, follows by
a mild variation on the arguments of Sections 2 and 3.
To obtain an additional equality with a total representation number, one should work with
the so-called composable algebra version of the LCH DGA, (Acomp, ∂), cf. [4, 9]. The underlying
algebra Acomp has generators b1, . . . , br and t±1
1 , . . . , t±1
` from Reeb chords and basepoints as well
as idempotent generators e1, . . . , ec corresponding to the components of K. Moreover, Acomp
has relations
eiej = δi,j ,
c∑
i=1
ei = 1,
eibk = δi,u(k)bk, bkei = δi,l(k)bk,
eitk = tkei = δi,s(k)tk, tkt
−1
k = t−1
k tk = es(k),
where the upper (resp. lower) endpoint of the Reeb chord bk is on the component Ku(k) (resp.
Kl(k)) and the basepoint tk sits on the Ks(k) component. Note that an algebra representation
f : Acomp → End(V ) is equivalent to a collection of vector spaces V1, . . . , Vc together with linear
maps
f(bk) : Vl(k) → Vu(k)
assigned to Reeb chords, and invertible linear maps
f(tk) : Vs(k) → Vs(k)
assigned to base points. Here, the Vi are determined from V via Vi = f(ei)V . (This is as
in the correspondence between quiver representations and representations of the corresponding
path algebra, see, e.g., [5, Section 1.2]. Except for the relation tkt
−1
k = t−1
k tk = es(k), the
composable algebra Acomp is precisely the path algebra associated to the quiver with vertices
indexed by components of K and edges corresponding to Reeb chords and base points.) The
DGA differential ∂ on Acomp is defined to satisfy ∂ei = ∂tk = 0 and by the usual holomorphic
disk count on Reeb chords.
For ~n = (n1, . . . , nc), with ni ≥ 1 as above, we denote by Rep1
(
K,
(
F~nq , 0
))
the set of DGA
representations f : (Acomp, ∂)→
(
End
(
⊕i Fniq
)
, 0
)
that, when viewed as quiver representations,
assign the collection of vector spaces Fn1
q , . . . ,Fncq to the components of K, i.e., f(ei) is the
projection to the Fniq component. Note that to obtain a Legendrian isotopy invariant we need
to adjust the normalizing factor from [20] used in Definition 4.1. This is done by defining the
(1-graded) total ~n-dimensional representation number of K to be
Rep1
(
K,F~nq
)
:=
∏
i,j
∣∣HomFq
(
Fnjq ,Fniq
)∣∣−rbi,j(K)/2
×
(∏
i
∣∣GL
(
n,Fniq
)∣∣−`i)∣∣Rep1
(
K,
(
F~nq , 0
))∣∣
32 J. Murray and D. Rutherford
=
∏
i,j
(
qninj
)−rbi,j(K)/2
×
∏
i
(
qni(ni−1)/2
ni∏
m=1
(
qm − 1
))−`i∣∣Rep1
(
K,
(
F~nq , 0
))∣∣,
where rbi,j(K) is the number of Reeb chords, bk, with u(k) = i and l(k) = j and `i is the number
of basepoints on the Ki component of K.
With the composable algebra used for K, a suitable modification of Theorem 6.1 from [20]
relating augmentations of the satellites S
(
K, ~β
)
with higher dimensional representations of
(Acomp(K), ∂) continues to hold. [The key point being that when components Ki and Kj are
satellited with ni and nj stranded braids respectively, each Reeb chord, bk, with u(k) = i and
l(k) = j corresponds to an ni × nj matrix of Reeb chords in the satellite S(K, ~β).] From this
starting point, the analog of the first equality of Theorem 1.1, Rep1
(
K,F~nq
)
= R1
~n,K(z), can be
deduced as in Section 4.
Acknowledgements
This article is dedicated to Dmitry Fuchs to whom the second author is grateful for his generosity
and support through years in grad school and beyond. Thank you Dmitry! DR acknowledges
support from Simons Foundation grant #429536.
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1 Introduction
2 The n-colored ungraded ruling polynomial
2.1 Legendrian fronts and ruling polynomials
2.2 A Legendrian BMW algebra
2.3 Legendrian satellites and reduced ruling polynomials
2.4 Inductive characterization of Rn,K1
3 Relation to the Kauffman polynomial
3.1 The Kauffman polynomial and the BMW algebra
3.2 Symmetrizer in BMWn and the n-colored Kauffman polynomial
3.3 Ruling polynomials via specializations of the Kauffman polynomial
3.4 Establishing (2) of Proposition 3.8
4 The n-colored ruling polynomial and representation numbers
4.1 Review of the Legendrian contact homology DGA
4.2 1-graded representation numbers
4.3 Strategy of the proof of Lemma 4.3
4.4 Four diagrams for the satellite, S(K, )
4.4.1 Diagrams for K and
4.4.2 Diagrams for Legendrian satellites
4.4.3 DGA generators
4.5 Representations and augmentations of the satellite
5 The multi-component case
References
|
| id | nasplib_isofts_kiev_ua-123456789-210593 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:17Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Murray, Justin Rutherford, Dan 2025-12-12T10:34:31Z 2020 Legendrian DGA Representations and the Colored Kauffman Polynomial. Justin Murray and Dan Rutherford. SIGMA 16 (2020), 017, 33 pages 1815-0659 2020 Mathematics Subject Classification: 53D42; 57M27 arXiv:1908.08978 https://nasplib.isofts.kiev.ua/handle/123456789/210593 https://doi.org/10.3842/SIGMA.2020.017 For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R¹ₙ, 𝛫(q) arises as a specialization 𝘍ₙ, 𝛫(a, q)∣ₐ⁻¹₌₀ of the n-colored Kauffman polynomial and (ii) when q is a power of two R¹ₙ, 𝛫(q) agrees with the total ungraded representation number, Rep₁(𝛫, 𝔽ⁿq), which is a normalized count of n-dimensional representations of (A(𝛫),∂) over the finite field 𝔽q. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m≠1. This article is dedicated to Dmitry Fuchs, to whom the second author is grateful for his generosity and support through the years in grad school and beyond. Thank you, Dmitry! DR acknowledges support from the Simons Foundation grant #429536. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Legendrian DGA Representations and the Colored Kauffman Polynomial Article published earlier |
| spellingShingle | Legendrian DGA Representations and the Colored Kauffman Polynomial Murray, Justin Rutherford, Dan |
| title | Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_full | Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_fullStr | Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_full_unstemmed | Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_short | Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_sort | legendrian dga representations and the colored kauffman polynomial |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210593 |
| work_keys_str_mv | AT murrayjustin legendriandgarepresentationsandthecoloredkauffmanpolynomial AT rutherforddan legendriandgarepresentationsandthecoloredkauffmanpolynomial |