Description:
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 ma 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.