The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

Abstract

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], 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[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].