Finite type invariants of cyclic branched covers
type: Item
instance of: scholarly article
title: Finite type invariants of cyclic branched covers
zbMATH Open document ID: 1081.57010
author: Andrew Kricker
author: Stavros Garoufalidis
published in: Topology
publication date: 2004-11-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0107220
review text: A knot defines a family of closed 3-manifolds indexed by integers, namely the cyclic branched coverings of the knot. It is a long standing question to find a formula for the Casson-Walker invariant of these manifolds. In this paper, the authors give a formula for this invariant in terms of the residues of a rational function (which measures the 2-loop part of the Kontsevich integral of the knot) and the signature function of the knot. The formula, obtained in the hypothesis that the cyclic cover is a rational homology sphere, reads \[ \lambda(\Sigma_{(M,K)}^p)=\frac{1}{3}\text{Res}_p^{t_1,t_2,t_3}Q(M,K)(t_1,t_2,t_3) +\frac{1}{8}\sigma_p(M,K). \] This formula is obtained as a consequence of another formula that computes the LMO invariant of the cyclic branched covers in terms of a rational invariant of the knot and its signature function.
reviewed by: Răzvan Gelca
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 2117931
zbMATH Keywords: signatures
zbMATH Keywords: rational lift of the Kontsevich integral
zbMATH Keywords: wheels
zbMATH Keywords: Casson-Walker invariant
MaRDI profile type: Publication
OpenAlex ID: W2029056285
arXiv ID: math/0107220
DOI: 10.1016/J.TOP.2001.12.001
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description: scientific article