Narrow Results

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.

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold $M$ can be obtained as the image of the Dijkgraaf–Witten invariant of $M$ by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and ${𝔽}_p$ in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and ${𝔽}_p$, and give a formula of them for $p=7$.