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2006•Journal of Knot Theory and Its RamificationsRequires access

THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

Alessia Cattabriga

Open publisher page 8 citations

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 S 2 ×S 1 , 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 S 3 . As corollaries some properties of the Alexander polynomial of knots in S 3 are extended to the case of (1,1)-knots in lens spaces.

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What this paper is about

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 S 2 ×S 1 , 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 S 3 . As corollaries some properties of the Alexander polynomial of knots in S 3 are extended to the case of (1,1)-knots in lens spaces.

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Available 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 S 2 ×S 1 , 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 S 3 . As corollaries some properties of the Alexander polynomial of knots in S 3 are extended to the case of (1,1)-knots in lens spaces.

Key concepts: Alexander polynomial, Mathematics, HOMFLY polynomial, Jones polynomial, Combinatorics, Knot polynomial, Lens space, Knot (papermaking)

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