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

TL;DR

This paper derives formulas for the colored Jones polynomial of double twist knots, revealing new q-hypergeometric series and a duality between the Kontsevich-Zagier series and unimodal sequence generating functions.

Contribution

It provides explicit formulas for the colored Jones polynomial of certain double twist knots and establishes a new duality relating these series to the Kontsevich-Zagier function.

Findings

New families of q-hypergeometric series generalizing the Kontsevich-Zagier series.

A duality at roots of unity between the Kontsevich-Zagier function and unimodal sequence generating functions.

Formulas for the colored Jones polynomial of 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.