The Colored Jones Polynomial and Kontsevich-Zagier Series for Double Twist Knots

View/ Open
Date
2017-10-20MFO Scientific Program
Research in Pairs 2016Series
Oberwolfach Preprints;2017,29Author
Lovejoy, Jeremy
Osburn, Robert
Metadata
Show full item recordOWP-2017-29
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.