Uniqueness of symmetries of knots
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.
scientific article; zbMATH DE number 3925805
scientific article