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

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K-(-m,K--p) and K-(-m,K-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,K-p) gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.