The 2-variable polynomial of cable knots



Morton, HR ORCID: 0000-0002-8524-2695 and Short, HB
(1987) The 2-variable polynomial of cable knots. Mathematical Proceedings of the Cambridge Philosophical Society, 101 (2). 267 - 278.

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Abstract

<jats:title>Abstract</jats:title><jats:p>The 2-variable polynomial <jats:italic>P<jats:sup>K</jats:sup></jats:italic> of a satellite <jats:italic>K</jats:italic> is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating <jats:italic>P</jats:italic> for 2-strand cables about them even though the knots themselves share the same <jats:italic>P</jats:italic>. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knot's polynomial <jats:italic>P</jats:italic>, are shown in certain cases to be detectable from the polynomial of a 2-cable about the knot.</jats:p>

Item Type: Article
Depositing User: Symplectic Admin
Date Deposited: 23 Jan 2017 10:06
Last Modified: 09 Aug 2021 11:09
DOI: 10.1017/s0305004100066627
URI: https://livrepository.liverpool.ac.uk/id/eprint/3005312