Title

The Finiteness Theorem for Symmetries of Knots and 3-Manifolds with Non-Trivial Characteristic Decompositions

Document Type

Article

Department

Mathematics (Pomona)

Publication Date

1986

Keywords

3-manifolds, knots, symmetries of knots

Abstract

In 1961, fox asked what knots have infinitely many symmetries. This question can actually be split into two questions. First, what knots have finite group actions of arbitrarily large orders? Second, what knots have infinitely many non-conjugate group actions of a given order? It was previously shown by the author that torus knots are the only knots with finite group actions of arbitrarily large orders. Now, we show that no non-trivial knots have infinitely many non-conjugate group actions of a given order. Thus we have completely answered the question posed by Fox.

In addition we show here that any 3-manifold with a non-trivial characteristic decomposition has only finitely many non-conjugate finite group actions. These results are in contrast to the fact that there exist examples of knot complements which have infinitely many non-homotopic finite cyclic actions of a given order.

Rights Information

© 1986 Elsevier B.V.

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