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Uniqueness of Free Actions on S3 Respecting a Knot

Published online by Cambridge University Press:  20 November 2018

Michel Boileau  and
Erica Flapan
Michel Boileau
Affiliation:
Université de Paris-Sud, Orsay, France
Erica Flapan
Affiliation:
Pomona College, Claremont, California
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In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.

THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 969 - 982
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Flapan, E., Infinitely periodic knots, Can. J. Math. 37 (1985), 17–28.Google Scholar
2. Hartley, R., Knots with free period, Can. J. Math. 37 (1981), 91–102.Google Scholar
3. Jaco, W. and Shalen, P., Seifert fibered spaces in 3-manifolds, Memoirs AMS (1979).CrossRefGoogle Scholar
4. Johannson, K., Homotopy equivalences of 3-manifolds with boundaries, Leeture Notes in Mathematics 761 (1979).CrossRefGoogle Scholar
5. Meeks, W. and Scott, P., Finite group actions on 3-manifolds, preprint.CrossRefGoogle Scholar
6. Meeks, W., Simon, L. and Yau, S.-T., Embedded minimal surfaces, exotic spheres and manifolds with positive ricci curvature, Ann. of Math. 116 (1982), 621–659.Google Scholar
7. Meeks, W. and Yau, S.-T., Topology of three dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980), 441–485.Google Scholar
8. Morgan, J. and Bass, H., The Smith conjecture (Academic Press, 1984).Google Scholar
9. Mostow, G., Strong rigidity for locally symmetric spaces, Ann. of Math. Study 78 (Princeton Univ. Press, 1973).Google Scholar
10. Sakuma, M., Uniqueness of symmetries of knots, preprint.Google Scholar
11. Schubert, H., Knoten und Vollringe, Acta Math 90 (1953), 131–286.Google Scholar
12. Smith, P. A., Transformations of finite period II, Ann. of Math. 40 (1939), 690–711.Google Scholar
13. Thurston, W., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357–381.Google Scholar
14. Waldhausen, F., Eine Klasse von 3-dimensionalen Mannigfaltigkeiten, Invent. Math. 3 (1967), 308–333, and 4 (1967), 87–117.Google Scholar
15. Waldhausen, F., On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56–88.Google Scholar