Torus knots and quantum modular forms

Kazuhiro Hikami, Jeremy Lovejoy

Research output: Contribution to journalArticle

10 Citations (Scopus)

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.

Original languageEnglish
Article number2
JournalResearch in Mathematical Sciences
Volume2
Issue number1
DOIs
Publication statusPublished - Dec 1 2015

Fingerprint

Torus knot
Q-series
Modular Forms
Colored Jones Polynomial
Left handed
Cyclotomic
Polynomials
Roots of Unity
Generating Function
Duality
Generalise
Series
Family

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics

Cite this

Torus knots and quantum modular forms. / Hikami, Kazuhiro; Lovejoy, Jeremy.

In: Research in Mathematical Sciences, Vol. 2, No. 1, 2, 01.12.2015.

Research output: Contribution to journalArticle

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