New York Journal of Mathematics
Volume 25 (2019), 1312-1349


Jeremy Lovejoy and Robert Osburn

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, II

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Published: November 29, 2019.
Keywords: double twist knots, colored Jones polynomial.
Subject: 33D15, 57M27.

Let K(m,p) denote the family of double twist knots where 2m-1 and 2p are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of K(-m,-p) and
K(-m,p). The latter case leads to new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. These generalized Kontsevich-Zagier series are conjectured to be quantum modular forms. We also use Bailey pairs and formulas of Walsh to find Habiro-type expansions for the colored Jones polynomials of K(m,p) and K(m,-p).


The authors would like to thank Paul Beirne and Katherine Walsh Hall for their helpful comments and suggestions. The second author would like to thank the Max-Planck-Institut für Mathematik for their support during the completion of this paper.

Author information

Jeremy Lovejoy:
Current Address:
Department of Mathematics
University of California at Berkeley
970 Evans Hall #3780, Berkeley, CA 94720-3840, USA
Permanent Address:
CNRS, Université Denis Diderot - Paris 7
Case 7014, 75205 Paris Cedex 13, FRANCE


Robert Osburn:
School of Mathematics and Statistics
University College Dublin
Belfield, Dublin 4, Ireland, and
Max-Planck-Institut für Mathematik
Vivatsgasse 7, D-53111, Bonn, Germany