Infinitely Periodic Knots

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S 0, S 1, or S 2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S 3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.