THE COLORED JONES FUNCTION AND ALEXANDER POLYNOMIAL FOR TORUS KNOTS

MORTON, HR ORCID: 0000-0002-8524-2695 (1995) THE COLORED JONES FUNCTION AND ALEXANDER POLYNOMIAL FOR TORUS KNOTS. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 117 (1). pp. 129-135. ISSN 0305-0041, 1469-8064

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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
Uncontrolled Keywords:	4902 Mathematical Physics, 4903 Numerical and Computational Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	23 Jan 2017 10:09
Last Modified:	06 Dec 2024 23:41
DOI:	10.1017/S0305004100072959
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3005308