  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.

Geometric Topology;Combinatorics;Number Theory