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

Abstract

Abstract Knot polynomials colored with symmetric representations of SLq(N) satisfy difference equations as functions of representation parameter, which look like quantization of classical $$ \mathcal{A} $$ A -polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum $$ \mathcal{C} $$ 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 = qN. Thus, the $$ \mathcal{C} $$ 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.