A 𝑞-series identity via the 𝔰𝔩₃ colored Jones polynomials for the (2,2𝔪)-torus link

Abstract

The colored Jones polynomial is a q q -polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A q q -series called a tail is obtained as the limit of the s l 2 \mathfrak {sl}_2 colored Jones polynomials { J n ( K ; q ) } n \{J_n(K;q)\}_n for some link K K , for example, an alternating link. For the s l 3 \mathfrak {sl}_3 colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the s l 3 \mathfrak {sl}_3 colored Jones polynomials colored by ( n , 0 ) (n,0) for the ( 2 , 2 m ) (2,2m) -torus link. These two expressions of the tail provide an identity of q q -series. This is a knot-theoretical generalization of the Andrews–Gordon identities for the Ramanujan false theta function.