Legendrian DGA Representations and the Colored Kauffman Polynomial

TL;DR

This paper establishes a connection between counts of ungraded Legendrian contact homology representations and the colored Kauffman polynomial, introducing an ungraded n-colored ruling polynomial that links algebraic and polynomial invariants of Legendrian knots.

Contribution

It introduces an ungraded n-colored ruling polynomial and relates it to the colored Kauffman polynomial and representation counts, extending previous work on Legendrian invariants.

Findings

The ungraded n-colored ruling polynomial arises as a specialization of the colored Kauffman polynomial.

When q is a power of two, the ruling polynomial matches the total ungraded representation number.

The work extends the relationship between polynomial invariants and Legendrian contact homology representations.

Abstract

For any Legendrian knot $K$ in standard contact ${\mathbb R}^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce an ungraded $n$-colored ruling polynomial, $R^1_{n,K}(q)$, as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) $R^1_{n,K}(q)$ arises as a specialization $F_{n,K}(a,q)\big|_{a^{-1}=0}$ of the $n$-colored Kauffman polynomial and (ii) when $q$ is a power of two $R^1_{n,K}(q)$ agrees with the total ungraded representation number, $\operatorname{Rep}_1\big(K, \mathbb{F}_q^n\big)$, which is a normalized count of $n$-dimensional representations of $(\mathcal{A}(K),\partial)$ over the finite field $\mathbb{F}_q$. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118, arXiv:1802.10531] concerning the colored HOMFLY-PT polynomial, $m$-graded representation numbers, and $m$-graded ruling polynomials with $m \neq 1$.