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A homological definition of the Jones polynomial

Stephen Bigelow

Geometry & Topology Monographs 4 (2002) 29–41

DOI: 10.2140/gtm.2002.4.29

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 β ∈ B2n. We define a covering space C of the space of unordered n–tuples of distinct points in the 2n–punctured disk. We then describe two n–manifolds S and T in C, and show that the Jones polynomial of L can be defined as an intersection pairing between S and βT. Our construction is similar to one given by Lawrence, but more concrete.

Keywords

Jones polynomial, braid group, plat closure, bridge position

Mathematical Subject Classification

Primary: 57M25

Secondary: 20F36, 57M27

References
Publication

Received: 30 November 2001
Revised: 4 April 2002
Accepted: 22 July 2002
Published: 19 September 2002

Authors
Stephen Bigelow
Department of Mathematics
University of California
Santa Barbara CA 93106
USA