Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 3, pages 29--41.

A homological definition of the Jones polynomial

Stephen Bigelow

Abstract. We give a new definition of the Jones polynomial. Let L be an oriented knot or link obtained as the plat closure of a braid beta in B_{2n}. We define a covering space tilde{C} of the space of unordered n-tuples of distinct points in the 2n-punctured disk. We then describe two n-manifolds tilde{S} and tilde{T} in tilde{C}, and show that the Jones polynomial of L can be defined as an intersection pairing between tilde{S} and beta tilde{T}. Our construction is similar to one given by Lawrence, but more concrete.

Keywords. Jones polynomial, braid group, plat closure, bridge position

AMS subject classification. Primary: 57M25. Secondary: 57M27, 20F36.

E-print: arXiv:math.GT/0201221

Submitted to GT on 30 November 2001. (Revised 4 April 2002.) Paper accepted 22 July 2002. Paper published 19 September 2002.

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Stephen Bigelow
Department of Mathematics and Statistics, University of Melbourne
Victoria 3010, Australia

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