On the Homology of Branched Cyclic Covers of Knots.

Abstract

We consider the sequence of finite branched cyclic covers of $S\sp3$ branched along a tame knot $K : S\sp1\to S\sp3$ and prove several results about the homology of these manifolds. 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 of the odd-fold covers and that contain the square roots of the orders of the even-fold covers quotiented by the order of the 2-fold cover (that these numbers are all integers follows from a theorem of Plans (P)). We also show that the $\doubz/p\sp{r}$-homology of this sequence of manifolds is periodic in every dimension, and we investigate these periods for the one-dimensional homology. Additionally, we give a new proof of Plans' theorem in the even-fold case (that the kernel of the map induced on homology by the covering projection $M\sb{k}\to M\sb2$ is a direct double for k even) in the style of Gordon's proof of Plans' theorem in the odd-fold case (that $H\sb1(M\sb{k}$) is a direct double for k odd).