TY - JOUR
T1 - Torus knots and quantum modular forms
AU - Hikami, Kazuhiro
AU - Lovejoy, Jeremy
N1 - Funding Information:
KH thanks S. Zwegers for discussions. He also thanks the organizers of ‘Lectures on q-Series and Modular Forms’ (KIAS, July 2013), ‘Modular Functions and Quadratic Forms - number theoretic delights’ (Osaka University, December 2013), ‘Low Dimensional Topology and Number Theory’ (MFO, August 2014). The work of KH is supported in part by JSPS KAKENHI Grant Number 23340115, 24654041, 26400079.
Publisher Copyright:
© 2015 Hikami and Lovejoy.
PY - 2015/12/1
Y1 - 2015/12/1
N2 - 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.
AB - 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.
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U2 - 10.1186/s40687-014-0016-3
DO - 10.1186/s40687-014-0016-3
M3 - Article
AN - SCOPUS:84975139492
SN - 2522-0144
VL - 2
JO - Research in Mathematical Sciences
JF - Research in Mathematical Sciences
IS - 1
M1 - 2
ER -