Geschlecht von Knoten mit zwei Brücken und die Faserbarkeit ihrer Außenräume
type: Item
instance of: scholarly article
title: Geschlecht von Knoten mit zwei Brücken und die Faserbarkeit ihrer Außenräume
zbMATH Open document ID: 0734.55001
DOI: 10.1007/BF01174564
author: Klaus Funcke
published in: Mathematische Zeitschrift
publication date: 1978-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/172640
review text: An algorithm is given for determining the genus of a knot k with two bridges and for deciding whether such a knot k has a Stallings fibration. One can characterize k by two odd numbers \(\alpha\) and \(\beta \in (- \alpha,\alpha).\) The author produces from such a pair a triple \((n,<n>,<n+1>)\), where n is [\(\alpha\) /\(\beta\) ], \(<n+1>=2(\alpha - n\beta),\) \(<n>=2\beta -<n+1>,\) and introduces a transformation of it into \((n,<n>,| <n+1>-2\lambda <n>|)\) if \(| <n+1>-2\lambda <n>| <<n><<n+1>,\) or into \((n,| <n>-2\lambda <n+1>|,<n+1>)\) if \(| <n>-2\lambda <n+1>| <<n+1><<n>.\) Here \(\lambda\) is uniquely defined. The iteration of this gives a sequence which ends with \(<n,2,0>\) or \(<n,0,2>\). The genus of the corresponding knot is one less in the first case and remains the same in the second one. Also the genus of k corresponding to (n,0,2) is n/2 and that corresponding to (n,2,0) is (n- 1)/2. The knot \(k(n,<n>,<n+1>)\) has a fibration exactly in case \(<n+1>><n>\) and all the \(\lambda\) in the sequence are equal to 1.
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4214936
zbMATH Keywords: genus
zbMATH Keywords: knot
zbMATH Keywords: Stallings fibration
MaRDI profile type: Publication
cites work: Q5331777
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description: scientific article; zbMATH DE number 4214936