Geodesic
Uniqueness of Free Actions on S 3 Respecting a Knot
Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 969-982

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In this paper we consider free actions of finite cyclic groups on the pair (S 3, K), where K is a knot in S 3. That is, we look at periodic diffeo-morphisms f of (S 3, 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, (S 3, 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 (S 3, K) are conjugate to elements of one cyclic group which acts freely on (S 3, 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. 
Boileau, Michel; Flapan, Erica. Uniqueness of Free Actions on S 3 Respecting a Knot. Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 969-982. doi: 10.4153/CJM-1987-049-3
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