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, $\beta \subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,\beta)}(z)$ 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, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of $K$. Conversely, for $m\neq 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to $\mathbb{F}q^n$, $\mbox{Rep}_m(K,\mathbb{F}_q^n)$, is exactly equal to $R^m{n,K}(q)$. In the case of $2$-graded representations, we show that $R^2_{n,K}=\mbox{Rep}_2(K, \mathbb{F}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.