Conway spheres as ideal points of the character variety
type: Item
instance of: scholarly article
title: Conway spheres as ideal points of the character variety
zbMATH Open document ID: 1264.57003
author: Luisa Paoluzzi
author: Joan Porti
published in: Mathematische Annalen
publication date: 2012-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.4673
review text: It was proved by \textit{M. Culler} and \textit{P. B. Shalen} in [Ann. Math. (2) 117, 109--146 (1983; Zbl 0529.57005)] that essential surfaces in \(3\)-manifolds are associated to ideal points of the character variety. However not all essential surfaces can be detected this way. In this paper, the authors prove that a special family of essential surfaces, namely Conway spheres, correspond to an ideal point of the character variety. More precisely, if \(L\) is a hyperbolic link admitting a unique Conway sphere \(C\) up to isotopy and if \(C'\) is the planar surface which is the intersection of \(C\) with the exterior of the link, then \(C'\) corresponds to an ideal point of the character variety of the exterior of \(L\). This result and its generalization to hyperbolic links admitting more than one Conway sphere is proved by studying properties of the pieces obtained by cutting the link exterior along Conway spheres and their character varieties and by exhibiting irreducible characters in the intersection of certain varieties which do not have a limit in the character variety of the link exterior.
reviewed by: Leila Ben Abdelghani
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6087535
zbMATH Keywords: character variety
zbMATH Keywords: Conway spheres
zbMATH Keywords: essential surface
zbMATH Keywords: ideal point
MaRDI profile type: Publication
OpenAlex ID: W2160267997
arXiv ID: 1008.4673
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DOI: 10.1007/S00208-011-0748-Y
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description: scientific article; zbMATH DE number 6087535