THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

Abstract

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3are extended to the case of (1,1)-knots in lens spaces.