Cusp structures of alternating links
type: Item
instance of: scholarly article
title: Cusp structures of alternating links
zbMATH Open document ID: 0810.57010
DOI: 10.1007/BF01232034
author: Iain R. Aitchison
author: E. Lumsden
author: J. Hyam Rubinstein
published in: Inventiones Mathematicae
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144028
review text: Given a finite, connected graph \(\Gamma\) in a 2-sphere in \(S^ 3\) there is naturally associated an alternating link \({\mathcal L}_ \Gamma\) whose projection to the 2-sphere meets each complementary region of \(\Gamma\) in its inscribed polygon. There is an associated decomposition of the complement \(S^ 3 - {\mathcal L}_ \Gamma\) into ideal polyhedra as introduced by Thurston in his notes on ``The Geometry and Topology of 3- manifolds''. This paper is a study of these constructions as tools in the understanding of the 3-manifolds \(S^ 3 - {\mathcal L}_ \Gamma\) and their Dehn fillings relative to questions such as the existence of incompressible surfaces, the existence of (possibly singular) metrics of negative curvature, determination by their fundamental group, and so on. Numerous examples of these phenomena are provided.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 220744
zbMATH Keywords: finite, connected graph in a 2-sphere
zbMATH Keywords: link complements
zbMATH Keywords: alternating link
zbMATH Keywords: ideal polyhedra
zbMATH Keywords: Dehn fillings
zbMATH Keywords: existence of incompressible surfaces
zbMATH Keywords: metrics of negative curvature
zbMATH Keywords: fundamental group
reviewed by: John Hempel
MaRDI profile type: Publication
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OpenAlex ID: W1970666683
description: scientific article; zbMATH DE number 220744