The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of $mathbb{F}_p$

Abstract

In 2004, W.D. 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, C.K. Zickert gave an analogue of the Neumann's work for free fields containing finite fields. The author formulated a simple method to calculate a weaker version of the Zickert's analogue, called the reduced Dijkgraaf-Witten invariant, for finite fields and gave a formula for twist knot complements and $mathbb{F}_p$ in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf-Witten invariants of double twist knot complements and $mathbb{F}_p$, and give a formula of them for p = 7.