The coloured Jones function and Alexander polynomial for torus knots



Morton, HR ORCID: 0000-0002-8524-2695
(1995) The coloured Jones function and Alexander polynomial for torus knots. Mathematical Proceedings of the Cambridge Philosophical Society, 117 (1). 129 - 135.

[img] Text
ColouredJonesAlexanderTorusKnot.pdf - Accepted Version

Download (233kB)

Abstract

<jats:title>Abstract</jats:title><jats:p>In [2] it was conjectured that the coloured Jones function of a framed knot <jats:italic>K</jats:italic>, or equivalently the Jones polynomials of all parallels of <jats:italic>K</jats:italic>, is sufficient to determine the Alexander polynomial of <jats:italic>K</jats:italic>. An explicit formula was proposed in terms of the power series expansion <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0305004100072959_inline1" />, where <jats:italic>J<jats:sub>K, k</jats:sub>(h)</jats:italic> is the <jats:italic>SU</jats:italic>(2)<jats:italic><jats:sub>q</jats:sub></jats:italic> quantum invariant of <jats:italic>K</jats:italic> when coloured by the irreducible module of dimension <jats:italic>k</jats:italic>, and <jats:italic>q = e<jats:sup>h</jats:sup></jats:italic> is the quantum group parameter.</jats:p><jats:p>In this paper I show that the explicit formula does give the Alexander polynomial when <jats:italic>K</jats:italic> is any torus knot.</jats:p>

Item Type: Article
Depositing User: Symplectic Admin
Date Deposited: 23 Jan 2017 10:09
Last Modified: 09 Aug 2021 11:09
DOI: 10.1017/s0305004100072959
URI: https://livrepository.liverpool.ac.uk/id/eprint/3005308