Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 5

Special issue on “Subfactors and Related Topics” in memory of Vaughan Jones

Guest Editors: Dietmar Bisch, Arthur Jaffe, Yasuyuki Kawahigashi, and Zhengwei Liu

The descendant colored Jones polynomials

Pages: 2307 – 2334

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n5.a2

Authors

Stavros Garoufalidis (International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, China)

Rinat Kashaev (Section de Mathématiques, Université de Genève, Switzerland)

Abstract

We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture.

Keywords

knots, Jones polynomial, colored Jones polynomials, Kashaev invariant, Habiro ring, Habiro polynomials, cyclotomic expansion, ADO invariants, descendants, holomorphic quantum modular forms, Volume Conjecture, Quantum Modularity Conjecture, $q$-holonomic functions, $q$-hypergeometric functions

Received 17 August 2021

Received revised 23 June 2022

Accepted 4 July 2022

Published 30 January 2024