An orientation for the SU(2)-representation space of knot groups
type: Item
instance of: scholarly article
title: An orientation for the SU(2)-representation space of knot groups
zbMATH Open document ID: 1019.57002
DOI: 10.1016/S0166-8641(02)00059-7
author: Michael Heusener
published in: Topology and its Applications
publication date: 2003-04-28T00:00:00Z
review text: Let \(K\subset S^3\) be a knot and denote by \(\hat R(K)\) the space of equivalence classes of irreducible representations of the knot group \(G=\pi_1(S^3\setminus K)\) in \(SU(2)\). Let \(Reg(K)\subset \hat R(K)\) be the space of regular representations, i.e irreducible representations \(\rho:G\rightarrow SU(2)\) such that \(H_\rho^1(G)\cong rb\) where \(H_\rho^*(G)=H^*(G,Ad\rho)\) denotes the twisted cohomology group of \(G\) with coefficients in \(su(2)\). It was proved by \textit{M. Heusener} and \textit{E. P. Klassen} in [Proc. Am. Math. Soc. 125, 3039-3047 (1997; Zbl 0883.57001)] that \(Reg(K)\subset \hat R(K)\) is a real one dimensional manifold. In this paper, the author proves that \(Reg(K)\) also carries a canonical orientation. A first consequence of this result is that there exists a large class of knots which have at least a one dimensional representation space \(\hat R(K)\). As a further application, the author explains a generalization of a result of \textit{X. S. Lin} [J. Differential Geometry 35, 337-357 (1992; Zbl 0774.57007)] \textit{M. Heusener} and \textit{J. Kroll} [Comment. Math. Helv. 73, 480-498 (1998; Zbl 0910.57004)]. Using the orientation of the representation space, he proves that the generalized Lin invariant and the signature of the knot are the same quantities as they turn out to be a weighted sum of zeros of the Alexander polynomial on the unit circle.
reviewed by: Leila Ben Abdelghani
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1901934
zbMATH Keywords: knot group
zbMATH Keywords: Casson invariant
zbMATH Keywords: representation spaces
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00059-7
OpenAlex ID: W2074972368
description: scientific article; zbMATH DE number 1901934