THE ALEXANDER POLYNOMIAL OF A TORUS KNOT WITH TWISTS



MORTON, HUGH R
(2006) THE ALEXANDER POLYNOMIAL OF A TORUS KNOT WITH TWISTS. Journal of Knot Theory and Its Ramifications, 15 (08). 1037 - 1047.

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Abstract

<jats:p> This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids [Formula: see text], where δ<jats:sub>p</jats:sub> = σ<jats:sub>p-1</jats:sub>σ<jats:sub>p-2</jats:sub> · σ<jats:sub>2</jats:sub>σ<jats:sub>1</jats:sub>, or equally of the q-string braids [Formula: see text]. As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| &gt; 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori. </jats:p>

Item Type: Article
Depositing User: Symplectic Admin
Date Deposited: 21 Apr 2016 10:00
Last Modified: 09 Jan 2021 08:32
DOI: 10.1142/s0218216506004920
URI: https://livrepository.liverpool.ac.uk/id/eprint/3000530