The skein polynomial of freely periodic knots (Q1612227)

The paper considers a question of Fox about knots which are invariant under a transformation of \(S^3\) of period \(p\). Murasugi and Traczyk have previously identified features of the Jones and Homfly polynomials of knots which are invariant under cyclic group actions with a fixed circle. The cases discussed in this paper are those where the knot is invariant under the free \((p,s)\)-lens space action of the cyclic group of order \(p\) and are termed \((p,s)\)-lens knots. Traczyk's techniques are adapted to prove results about the coefficients of the Homfly polynomial of a \((p,s)\)-lens knot in terms firstly of \((p,s)\)-closed braids and then of suitable torus knots. As an illustration of the general results it is shown that sufficient information is available to analyse most 9-crossing knots in the cases \(p=3\) or \(5\).