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, which we call the n-cyclic polynomial. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S^2\times S^1, a result obtained by J. Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some properties of the Alexander polynomial of knots in the 3-sphere are extended to the case of (1,1)-knots in lens spaces.
Keywords
Cite
@article{arxiv.math/0501394,
title = {The Alexander polynomial of (1,1)-knots},
author = {Alessia Cattabriga},
journal= {arXiv preprint arXiv:math/0501394},
year = {2007}
}
Comments
11 pages, 1 figure. A corollary has been extended, and a new example added. Accepted for publication on J. Knot Theory Ramifications