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

Abstract

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.

24 pages, added dedication to Toshie Takata, to appear in the Journal of Knot Theory and its Ramifications

References in corpus (7)

• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Difference equation of the colored Jones polynomial for torus knot
• Irreducibility of q-difference operators and the knot 7_4
• Large color $R$-matrix for knot complements and strange identities
• Quantum modularity of partial theta series with periodic coefficients
• Asymptotics of the Colored Jones Polynomial and the A-Polynomial
• Congruences for Fishburn numbers modulo prime powers

Cited by in corpus (3)

• Large color $R$-matrix for knot complements and strange identities
• Generalized Fishburn numbers and torus knots
• On knots, complements, and 6j-symbols