Narrow Results

Let $M$ be a closed oriented 3-manifold and let $G$ be a discrete group. We consider a representation $\rho :{\pi }_1(M)\to G$. For a 3-cocycle $\alpha$, the Dijkgraaf–Witten invariant is given by $({\rho }^{\ast }\alpha )[M]$, where ${\rho }^{\ast }:H^3(G)\to H^3(M)$ is the map induced by $\rho$, and $[M]$ denotes the fundamental class of $M$. Note that $({\rho }^{\ast }\alpha )[M]=\alpha ({\rho }_{\ast }[M])$, where ${\rho }_{\ast }:H_3(M)\to H_3(G)$ is the map induced by $\rho$, we consider an equivalent invariant ${\rho }_{\ast }[M]\in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\mathrm{\text{SL}}}_2ℂ$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2ℂ)$ to the Bloch group of $ℂ$.

In this paper, by replacing $ℂ$ with a finite field ${𝔽}_p$, we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL${}_2{𝔽}_p$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2{𝔽}_p)$ to the Bloch group of ${𝔽}_p$.

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 $p=7$.