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

Abstract

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

21 pages

References in corpus (6)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Equations on knot polynomials and 3d/5d duality
• Perspectives of differential expansion
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Large Nc gauge theory with quarks in high representations

Cited by in corpus (4)

• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Defect and degree of the Alexander polynomial
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket