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The coloured Jones function and Alexander polynomial for torus knots

  • H. R. Morton (a1)
Abstract
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 , 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.

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[4] M. Rosso and V. F. R. Jones . On the invariants of torus knots derived from quantum groups, Journal of Knot Theory and its Ramifications 2 (1993), 97112.

[5] N. Y. Reshetikhin and V. G. Turaev . Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), 126.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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