The Alexander polynomial of a torus knot with twists

Morton, Hugh R ORCID: 0000-0002-8524-2695 (2006) The Alexander polynomial of a torus knot with twists JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, 15 (8). pp. 1037-1047. ISSN 0218-2165, 1793-6527

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Abstract

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 (δ<inf>p</inf>)^qσ<inf>1</inf>^2m, where δ<inf>p</inf> = σ<inf>p-1</inf>σ <inf>p-2</inf>·σ<inf>2</inf>σ<inf>1</inf>, or equally of the q-string braids (δ<inf>q</inf>)^pσ<inf>1</inf> ^2m. 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| > 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. © World Scientific Publishing Company.

Item Type:	Article
Additional Information:	## TULIP Type: Articles/Papers (Journal) ##
Uncontrolled Keywords:	torus knot, twist, Dehn surgery, multi-variable Alexander polynomial
Depositing User:	Symplectic Admin
Date Deposited:	21 Apr 2016 10:00
Last Modified:	16 Jun 2026 10:58
DOI:	10.1142/S0218216506004920
Related Websites:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3000530
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