Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The coloured Jones function and Alexander polynomial for torus knots

H. R. Mortona1

a1 Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool, L69 3BX, U.K.

Abstract

In [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion S0305004100072959_inline1, where JK, k(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = eh is the quantum group parameter.

In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

(Received July 05 1993)