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Proceedings of the American Mathematical Society

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


Author: Wataru Yuasa
Journal: Proc. Amer. Math. Soc. 146 (2018), 3153-3166
MSC (2010): Primary 57M27; Secondary 11P84, 05A30
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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Additional Information

Wataru Yuasa
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Email: yuasa.w.aa@m.titech.ac.jp

DOI: https://doi.org/10.1090/proc/13907
Received by editor(s): December 21, 2016
Received by editor(s) in revised form: July 16, 2017, and July 24, 2017
Published electronically: March 20, 2018
Communicated by: David Futer
Article copyright: © Copyright 2018 American Mathematical Society

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