Uniqueness of symmetries of knots
type: Item
instance of: scholarly article
title: Uniqueness of symmetries of knots
zbMATH Open document ID: 0578.57003
DOI: 10.1007/BF01179425
author: Makoto Sakuma
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173731
review text: A symmetry of a knot K in \(S^ 3\) is a finite (cyclic) group action on \((S^ 3,K)\) which preserves the orientations of both \(S^ 3\) and K. The following problems are considered: (a) Are the symmetries of K of a given order unique? (b) More strongly, does K admit a ''universal symmetry''? The author gives certain sufficient conditions for (a) and (b) to hold. Further by examples, the conditions are seen to be the ''best possible''. It should be noted that there is a knot, for which (a) is affirmative but (b) is negative. For semi-free symmetries, the following results are obtained: (1) A free symmetry of a prime knot is uniquely determined by its order. (2) A symmetry of a knot which realizes a cyclic period n is unique, if \(n\geq 3\). (3) For any integer s, there is a knot which admits more than s inequivalent symmetries which realize cyclic period 2. The result (1) was also obtained by \textit{M. Boileau} and \textit{E. Flapan} [preprint], independently.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 3925805
zbMATH Keywords: symmetry of a knot
zbMATH Keywords: universal symmetry
zbMATH Keywords: semi-free symmetries
zbMATH Keywords: prime knot
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3925805