Algebra of quantum ${\cal C}$-polynomials

TL;DR

This paper introduces quantum ${ m C}$-polynomials, a new algebraic framework for understanding knot polynomials, providing finite difference equations for their coefficients, which simplifies the analysis of symmetric representation invariants.

Contribution

It develops the theory of quantum ${ m C}$-polynomials, deriving finite order difference equations for knot polynomial coefficients, expanding the algebraic structure beyond classical ${ m A}$-polynomials.

Findings

Derived two finite order difference equations for knot polynomial coefficients.

Showed that ${ m C}$-polynomials form an algebraic ring.

Applied the framework to defect zero knots, especially twist families.

Abstract

Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions of representation parameter, which look like quantization of classical ${\cal A}$-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum ${\cal C}$-polynomials. It turns out that, for each knot, one can actually derive two difference equations of a finite order for these coefficients, those with shifts in spin $n$ of the representation and in $A=q^N$. Thus, the ${\cal C}$-polynomials are much richer and form an entire ring. We demonstrate this with the examples of various defect zero knots, mostly discussing the entire twist family.