The Alexander polynomial of (1,1)-knots

Alessia Cattabriga

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The Alexander polynomial of (1,1)-knots

Alessia Cattabriga

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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.

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Alessia Cattabriga. 2005-10-21. The Alexander polynomial of (1,1)-knots. https://arxiv.org/abs/math/0501394

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