Once-punctured Klein bottles in knot complements
type: Item
instance of: scholarly article
title: Once-punctured Klein bottles in knot complements
zbMATH Open document ID: 1063.57008
author: Enrique Ramírez-Losada
author: Luis Valdez-Sanchez
published in: Topology and its Applications
publication date: 2005-02-22T00:00:00Z
review text: For a knot \(K\) in the \(3\)-sphere \(S^3\), a Seifert Klein bottle of \(K\) is a once-punctured Klein bottle \(P\) with \(\partial P=K\). If \(K\) is non-trivial and not a \(2\)-cable, then \(F\) is essential. That is, the restriction of \(F\) to the knot exterior \(E(K)\) is incompressible and boundary-incompressible. Furthermore, it was shown by \textit{K.\ Ichihara, M.\ Ohtouge} and \textit{M.\ Teragaito} [Topology Appl. 122, No. 3, 467--478 (2002; Zbl 0997.57009)] that if \(K\) admits two Seifert Klein bottles with different boundary slopes then \(K\) admits just two such slopes and their slopes are at a distance \(4\) or \(8\) from each other, and if at distance \(8\) then \(K\) is the figure-eight knot. Also, they conjectured that if at distance \(4\) then \(K\) must belong to a certain family of knots. The paper under review gives an affirmative solution to this conjecture. As a corollary, the authors obtain that the figure-eight knot and the \((-2,3,7)\)-pretzel knot (and its mirror image) are the only hyperbolic knots with more than one Seifert Klein bottle boundary slope. The argument is the analysis of the graph pair coming from intersection between two Seifert Klein bottles. Also, there is a discussion in a general situation. For an irreducible \(3\)-manifold \(M\) with an incompressible torus boundary component \(T\), consider the case that \(M\) contains two essential once-punctured Klein bottles whose boundaries lie on \(T\). Then their boundary slopes are shown to be at a distance at most \(8\). Moreover, if the distance is \(6\) or \(8\), then the manifold \(M\) is determined.
reviewed by: Masakazu Teragaito
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 2136462
zbMATH Keywords: knot
zbMATH Keywords: Klein bottle
zbMATH Keywords: boundary slope
MaRDI profile type: Publication
full work available at URL: j.topol.2003.02.006
OpenAlex ID: W1985777817
cites work: The knots in \(D^ 2\times S^ 1\) which have nontrivial Dehn surgeries that yield \(D^ 2\times S^ 1\)
cites work: Boundary slopes of punctured tori in 3-manifolds
cites work: Toroidal and boundary-reducing Dehn fillings
cites work: Dehn surgery on knots in solid tori creating essential annuli
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DOI: 10.1016/J.TOPOL.2003.02.006
description: scientific article; zbMATH DE number 2136462