On the reduced Dijkgraaf-Witten invariant of knots in the Bloch group of Fp – Hiroaki Karuo (RIMS, Kyoto )


Sala Seminari (Dip. Matematica).


For a closed oriented 3-manifold $M$, a discrete group $G$, a 3-cocycle $\alpha$ of $G$, and a representation $\rho \colon \pi_1(M) \to G$, the Dijkgraaf–Witten invariant is defined to be $\rho^\ast \alpha [M]$, where $[M]$ is the fundamental class of $M$, and $\rho^\ast \alpha$ is the pull-back of $\alpha$ 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 hyperbolic volume and the Chern–Simons invariant of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\rm SL}_2 \Bbb{C}$ by the Bloch–Wigner map $H_3(M)\to \mathcal{B}(\Bbb{C})$, where $\mathcal{B}(\Bbb{C})$ is the Bloch group of $\Bbb{C}$. Further, in 2013, Hutchinson gave a construction of the Bloch–Wigner map $H_3({\rm SL}_2 \Bbb{F}_p)\to \mathcal{B} (\Bbb{F}_p)$ explicitly, where $p$ is prime, and $\Bbb{F}_p$ is the finite field of order $p$. In this talk, I calculate the reduced Dijkgraaf–Witten invariant of the complement of knots, especially twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL$_2\Bbb{F}_p$ by the Bloch–Wigner map $H_3({\rm SL}_2\Bbb{F}_p) \to \mathcal{B}(\Bbb{F}_p)$.

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