Torus knots and quantum modular forms

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• 作者：Kazuhiro Hikami ; Jeremy Lovejoy
• 关键词：Torus knots ; Colored Jones polynomial ; Bailey pairs ; Quantum modular forms
• 刊名：Research in the Mathematical Sciences
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2
• 期：1
• 全文大小：483KB
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• 作者单位：Kazuhiro Hikami (1)
  Jeremy Lovejoy (2)
  
  1. Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan
  2. CNRS, LIAFA, Universite Denis Diderot - Paris 7, Case 7014, Paris Cedex 13, 75205, France
  
• 刊物类别：Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis;
• 刊物主题：Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer International Publishing
• ISSN：2197-9847

文摘

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. 2010 MSC:11F37; 33D15; 57M27 Keywords Torus knots Colored Jones polynomial Bailey pairs Quantum modular forms