Uniqueness of Free Actions on Ssp3 Respecting a Knot

Document Type

Article

Department

Mathematics (Pomona)

Publication Date

1987

Keywords

free actions, knot, cyclic group

Abstract

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 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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