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.
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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| > 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 |
|---|---|
| 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: | 15 Dec 2022 07:32 |
| DOI: | 10.1142/S0218216506004920 |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3000530 |
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