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 , \operatorname{Rep}_m(K, \mathbb{F}_q^n) , is exactly equal to R^m_{n,K}(q) . In the case of 2 \nbdash graded representations, we show that R^2_{n,K}(q)=\mathrm {Rep}_2(K, \mathbb{F}_q^n) arises as a specialization of the n -colored HOMFLY-PT polynomial.