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.

中文翻译：

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卫星规则多项式、DGA 表示和彩色 HOMFLY-PT 多项式

我们在 $\mathbb{R}^3$ 中建立了两类 Legendrian 结不变量之间的关系：Chekanov-Eliashberg DGA 和卫星规则多项式的表示数。对于正排列辫子，$\beta \subset J^1S^1$，我们给出了 $m$ 分级规则多项式 $R^m_{S(K,\beta)}( z)$ 的 $K$ 卫星的 $\beta$ 专用于 $z=q^{1/2}-q^{-1/2}$，$q$ 为素幂，我们使用这个公式证明任意 $m$-graded 卫星判定多项式 $R^m_{S(K,L)}$ 是由 $K$ 的 Chekanov-Eliashberg DGA 确定的。相反，对于 $m\neq 1$，我们引入了 $n$-colored $m$-graded 规则多项式 $R^m_{n,K}(q)$，与 $n$-colored 严格类比HOMFLY-PT多项式，并证明$K$到$\mathbb{F}_q^n$, $\mbox{Rep}_m(K,\mathbb{F}_q^ n)$，正好等于 $R^m_{n,K}(q)$。在 $2$-graded 表示的情况下，我们表明 $R^2_{n,K}=\mbox{Rep}_2(K, \mathbb{F}_q^n)$ 作为 $n 的特化出现$-colored HOMFLY-PT 多项式。