Torus knots and quantum modular forms

By Kazuhiro Hikami, Jeremy Lovejoy. 2014

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AbstractIn this paper we compute a q-hypergeometric expression for the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot (2,2t+1). We use this to define a family of q-series, the simplest case of which is the generating function for strongly unimodal sequences. Special cases of these q-series are quantum modular forms, and at roots of unity, these are dual to the generalized Kontsevich-Zagier series introduced by the first author. This duality generalizes a result of Bryson, Pitman, Ono, and Rhoades. We also compute Hecke-type expansions for our family of q-series. 2010 MSC:11F37; 33D15; 57M27

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c9a5096ed7d64df598df545b14eaf8441d31c6ff