Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial

Abstract

We establish relationships between two classes of invariants of Legendrian knots in ${\mathbb{R}}^3$: representation numbers of the Chekanov–Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $\beta \subset J^1{S}^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R_{S(K,\beta )}^m(z)$ of the satellite of $K$ with $\beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power, and we use this formula to prove that arbitrary $m$-graded satellite ruling polynomials, $R_{S(K,L)}^m$, are determined by the Chekanov–Eliashberg DGA of $K$. Conversely, for $m\ne 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R_{n,K}^m(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to ${\mathbb{F}}_q^n$, ${\mathrm{Rep}}_m(K,{\mathbb{F}}_q^n)$, is exactly equal to $R_{n,K}^m(q)$. In the case of $2$\nbdash graded representations, we show that $R_{n,K}^2(q)={\mathrm{Rep}}_2(K,{\mathbb{F}}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.