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.
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