#### 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.

Notes on file formats
Stephen Bigelow

Department of Mathematics and Statistics, University of Melbourne

Victoria 3010, Australia

Email: bigelow@unimelb.edu.au

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