Uniqueness of Free Actions on Ssp3 Respecting a Knot

Document Type



Mathematics (Pomona)

Publication Date



free actions, knot, cyclic group


In this paper we consider free actions of finite cyclic groups on the pair (S³, K), where K is a knot in . That is, we look at periodic diffeomorphisms 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 diffeomorphisms 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.

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© 1987 Canadian Mathematical Society

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