Let M be a closed oriented 3-manifold and let G be a discrete group. We consider a representation ρ:π1(M) → G. For a 3-cocycle α, the Dijkgraaf–Witten invariant is given by (ρ∗α)[M], where ρ∗:H3(G) → H3(M) is the map induced by ρ, and [M] denotes the fundamental class of M. Note that (ρ∗α)[M] = α(ρ ∗[M]), where ρ∗:H3(M) → H3(G) is the map induced by ρ, we consider an equivalent invariant ρ∗[M] ∈ H3(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 = SL2ℂ by the Bloch–Wigner map from H3(SL2ℂ) to the Bloch group of ℂ. In this paper, by replacing ℂ with a finite field 𝔽p, 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 SL2𝔽p by the Bloch–Wigner map from H3(SL2𝔽p) to the Bloch group of 𝔽p.

中文翻译：

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有限域 Bloch 群中扭结的约化 Dijkgraaf-Witten 不变量

让米是一个封闭的 3 流形并让G成为离散群。我们考虑一个表示ρ：π1(米) → G. 对于 3-cocycleα, Dijkgraaf-Witten 不变量由下式给出(ρ*α)[米]， 在哪里ρ*：H3(G) → H3(米)是由ρ， 和[米]表示基本类米. 注意(ρ*α)[米] = α(ρ *[米])， 在哪里ρ*：H3(米) → H3(G)是由ρ, 我们考虑一个等价的不变量ρ*[米] ∈ H3(G), 我们也认为它是 Dijkgraaf-Witten 不变量。2004 年，Neumann 描述了复双曲体积米就 Dijkgraaf-Witten 不变量的图像而言G = SL2ℂ由 Bloch-Wigner 地图从H3(SL2ℂ)布洛赫组ℂ. 在本文中，通过替换ℂ有一个有限域𝔽p，我们计算了扭结补的简化的 Dijkgraaf-Witten 不变量，其中简化的 Dijkgraaf-Witten 不变量是 SL 的 Dijkgraaf-Witten 不变量的图像2𝔽p由 Bloch-Wigner 地图从H3(SL2𝔽p)布洛赫组𝔽p.