A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link

Abstract

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the ${\mathfrak{sl}}_2$ colored Jones polynomials $\{J_n(K;q){\}}_n$ for some link $K$, for example, an alternating link. For the ${\mathfrak{sl}}_3$ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the ${\mathfrak{sl}}_3$ colored Jones polynomials colored by $(n,0)$ for the $(2,2m)$-torus link. These two expressions of the tail provide an identity of $q$-series. This is a knot-theoretical generalization of the Andrews-Gordon identities for the Ramanujan false theta function.

11 pages, many TikZ pictures, 1 table

References in corpus (3)

• Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial
• Rogers--Ramanujan computer searches
• Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathfrak{sl}}_3$ false theta functions

Cited by in corpus (3)

• Coloured ${\mathfrak{sl}}_r$ invariants of torus knots and characters of ${\mathcal{W}}_r$ algebras
• $A_2$ Skein Representations of Pure Braid Groups
• The zero stability for the one-row colored ${\mathfrak{sl}}_3$ Jones polynomial