Hecke-type formulas for families of unified Witten–Reshetikhin–Turaev invariants

Every closed orientable $3$-manifold can be constructed by surgery on a link in $S^3$. In the case of surgery along a torus knot, one obtains a Seifert fibered manifold. In this paper we consider three families of such manifolds and study their unified Witten–Reshetikhin–Turaev (WRT) invariants. Thanks to recent computation of the coefficients in the cyclotomic expansion of the colored Jones polynomial for $(2, 2t + 1)$-torus knots, these WRT invariants can be neatly expressed as $q$-hypergeometric series which converge inside the unit disk. Using the Rosso–Jones formula and some rather non-standard techniques for Bailey pairs, we find Hecke-type formulas for these invariants. We also comment on their mock and quantum modularity.

Keywords

unified WRT invariants, torus knots, integral homology spheres, Bailey pairs, quantum modular forms, mock theta functions

2010 Mathematics Subject Classification

33D15, 57M27

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Received 23 October 2016

Published 14 August 2017