Companionship of knots and the smith conjecture

Abstract

This paper studies the Smith Conjecture in terms of H. Schubert’s theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of Zₚ on S³. Theorem. Every essential torus in an invariant knot space C(J) of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in C(J), this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theo­rem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for ail simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.