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

Yuasa, Wataru

doi:10.1090/proc/13907

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

Proc. Amer. Math. Soc. 146 (2018), 3153-3166

DOI: https://doi.org/10.1090/proc/13907

Published electronically: March 20, 2018

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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.

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