On the knot complement problem for non-hyperbolic knots
type: Item
instance of: scholarly article
title: On the knot complement problem for non-hyperbolic knots
zbMATH Open document ID: 1194.57022
author: Daniel Matignon
published in: Topology and its Applications
publication date: 2010-06-22T00:00:00Z
review text: The paper under review introduces two infinite families of knots in lens spaces. One of them consists of knots whose exteriors are Seifert fibered manifolds, and the other consists of knots whose exteriors contain essential tori. Each knot of those families admits a non-trivial Dehn surgery which creates a lens space homeomorphic to the original lens space, called a cosmetic surgery in the literature, under an orientation-reversing homeomorphism. Such a surgery is shown to be unique. No two knots of each family have homeomorphic exteriors. The main point is that these families give all non-hyperbolic non-trivial knots in lens spaces which admit cosmetic surgeries. The construction of the families is concrete. On the other hand, \textit{S. A. Bleiler, C. D. Hodgson} and \textit{J. R. Weeks} [Geom. Topol. Monogr. 2, 23--34 (1999; Zbl 0948.57017)] gave a single example of a hyperbolic knot in a lens space which admits a cosmetic surgery. The author of the current paper announces an infinite family of such hyperbolic knots. As a corollary, non-hyperbolic knots in lens spaces, more generally, closed atoroidal irreducible Seifert fibered manifolds, are determined by their complements, except for the axes in lens spaces \(L(p,q)\) with \(q^2\equiv \pm 1\pmod{p}\).
reviewed by: Masakazu Teragaito
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5724585
zbMATH Keywords: cosmetic surgery
zbMATH Keywords: Dehn surgery
zbMATH Keywords: lens space
zbMATH Keywords: knot complement problem
describes a project that uses: SnapPea
MaRDI profile type: Publication
full work available at URL: j.topol.2010.03.009
OpenAlex ID: W1982625436
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cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
DOI: 10.1016/J.TOPOL.2010.03.009
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description: scientific article; zbMATH DE number 5724585