TY - JOUR

T1 - Hecke-type formulas for families of unified Witten-Reshetikhin-Turaev invariants

AU - Hikami, Kazuhiro

AU - Lovejoy, Jeremy

N1 - Funding Information:
We would like to thank Kathrin Bringmann and Larry Rolen for helpful discussions on indefinite ternary theta functions. The work of the first author is supported by JSPS KAKENHI Grant Number JP16H03927, JP16H02143

PY - 2017

Y1 - 2017

N2 - Every closed orientable 3-manifold can be constructed by surgery on a link in S3. 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.

AB - Every closed orientable 3-manifold can be constructed by surgery on a link in S3. 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.

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U2 - 10.4310/CNTP.2017.v11.n2.a1

DO - 10.4310/CNTP.2017.v11.n2.a1

M3 - Article

AN - SCOPUS:85027499575

SN - 1931-4523

VL - 11

SP - 249

EP - 272

JO - Communications in Number Theory and Physics

JF - Communications in Number Theory and Physics

IS - 2

ER -