The Alexander polynomial for closed braids in lens spaces

For a knot K in a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold of p/q-surgery along K. We say that the manifold or the surgery is of lens type if Σ(K;p/q) has the same Reidemeister torsion as a lens space.We prove that, for Σ(K;p/q) to be of lens type, it is a necessary and sufficient condition that the Alexander polynomial ΔK(t) of K is equal to that of an (i, j)-torus knot T(i, j) modulo (tp – 1).We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion as L(p, q') then (1) (2) The multiple of ΣK(tk) over k ∈ (i) is ±tm modulo (tp – 1), where (i) is the subgroup in (Z/pZ)×/{±1} generated by i. Conversely, if a subgroup H of (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k ∈ H) is ±tm modulo (tp – 1), then H includes i or j.Here, i, j are the parameters of the torus knot above.

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The Alexander polynomial of links in lens spaces

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500494 ◽

2019 ◽

Vol 28 (08) ◽

pp. 1950049 ◽

Cited By ~ 2

Author(s):

E. Horvat ◽

Boštjan Gabrovšek

Keyword(s):

Lens Space ◽

Alexander Polynomial ◽

Lens Spaces ◽

Cutting Out ◽

Skein Relation

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fiber. It follows from this relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.

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THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216506005019 ◽

2006 ◽

Vol 15 (09) ◽

pp. 1119-1129 ◽

Cited By ~ 2

Author(s):

A. CATTABRIGA

Keyword(s):

Fundamental Group ◽

Alexander Polynomial ◽

Lens Spaces ◽

Cyclic Covering ◽

Genus One

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 S3 are extended to the case of (1,1)-knots in lens spaces.

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