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

Abstract

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 A $$ \mathcal{A} $$ -polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum C $$ \mathcal{C} $$ -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 C $$ \mathcal{C} $$ -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.

Keywords

• Chern-Simons Theories
• Wilson, ’t Hooft and Polyakov loops
• Topological Field Theories