Knots with the lens space surgery
type: Item
instance of: scholarly article
title: Knots with the lens space surgery
zbMATH Open document ID: 1278.57005
published in: Mediterranean Journal of Mathematics
publication date: 2013-03-20T00:00:00Z
review text: In this paper a family of knots \(K_{m,n}\) is defined, where \(m\), \(n\) are positive integers. It is shown that each knot in this family is a hyperbolic strongly invertible \((1,1)\)-knot. An explicit presentation of the fundamental group of each of these knots is given, and then an explicit presentation is given of the fundamental group of the manifolds obtained by Dehn surgery on these knots. In the case that \(n=m+3\), it is shown that \(K_{m,m+3}\) has some exceptional Dehn surgeries, namely, it is shown that surgery with slopes \(9m+11\) and \(9m+7\) produces manifolds containing an incompressible torus but not containing a Klein bottle, surgery with slopes \(9m+6\) and \(9m+10\) produces Seifert fibered manifolds, while surgery with slope \(9m+8\) produces a lens space. This is done by a careful manipulation of a presentation of the corresponding fundamental groups. So, these knots have two toroidal Dehn surgeries at distance 4. However, this result gives a different description of a family of knots that was already known. \textit{C. McA. Gordon} and \textit{Y.-Q. Wu} have determined all hyperbolic knots with a pair of toroidal surgeries at distance 4 [Mem. Am. Math. Soc. 909, 140 p. (2008; Zbl 1166.57014)], and it follows that the knot \(K_{m,m+3}\) corresponds to the knot \(L_2(-m-1)\) as defined by Gordon and Wu.
reviewed by: Mario Eudave-Muñoz
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 6147205
zbMATH Keywords: knot
zbMATH Keywords: Dehn surgery
zbMATH Keywords: exceptional surgery
zbMATH Keywords: toroidal manifold
zbMATH Keywords: lens space
zbMATH Keywords: Seifert fibered space
zbMATH Keywords: fundamental group
author: Alberto Cavicchioli
author: A. I. Telloni
describes a project that uses: SnapPea
MaRDI profile type: Publication
full work available at URL: s00009-011-0163-3
OpenAlex ID: W2061756425
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DOI: 10.1007/S00009-011-0163-3
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description: scientific article; zbMATH DE number 6147205