RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

Cyclic group actions and knots

Using algebraic techniques derived partly from results on logarithmic forms, the existence of infinitely many inequivalent cyclic group actions fixing a given knot is investigated, together with the related problem of whether infinitely many distinct knots can arise as the branched cyclic covers of a given knot.