Torus knots and quantum modular forms

Abstract

In 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.