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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

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