On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces
type: Item
instance of: scholarly article
title: On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces
zbMATH Open document ID: 0616.57008
DOI: 10.1007/BF01456287
author: Michel Boileau
author: Markus Rost
author: Heiner Zieschang
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164340
review text: A Heegaard decomposition of the exterior E of a knot k is the union of a handlebody \(H_ g\) of genus g with (g-1) 2-handles which are attached along curves on \(\partial H_ g\). The Heegaard genus of k is the minimal genus of decompositions of E. In this paper, the Heegaard decompositions of genus 2 of torus knot exteriors are classified with respect to homeomorphisms. It turns out that, in general, there are three different classes which are also the isotropy classes. A similar result is obtained for Seifert fibre spaces having a disc as base and two exceptional fibres, such as the torus knot exteriors have. The proof of these classification theorems are based on two results: (1) The classification of the Nielsen equivalence classes of one relator presentations of the fundamental groups, \(<S,T | S^ pT^{-q}>\), of these manifolds which has been done by the third author and D. J. Collins. (2) The determination of the ''geometric'' one-relator presentations of the fundamental group of these manifolds. These are the presentations obtained from a genus 2 Heegaard decomposition. Moreover, the different Heegaard decompositions of genus 2 of the manifolds are geometrically described.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 3998785
zbMATH Keywords: Heegaard decomposition
zbMATH Keywords: exterior of a knot
zbMATH Keywords: Heegaard decompositions of genus 2 of torus knot exteriors
zbMATH Keywords: Seifert fibre spaces
zbMATH Keywords: Nielsen equivalence classes of one relator presentations of the fundamental groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3998785