A homological definition of the Jones polynomial

2002/01/31 by Stephen Bigelow
Mathematics · #math.GT #msc:57M25 #msc:57M27 #msc:20F36

paper · pdf

published as Geom. Topol. Monogr. 4 (2002) 29-41 · Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper3.abs.html

arxiv created 2002/11/15 · arxiv updated 2009/11/30

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 tildeC of the space of unordered n-tuples of distinct points in the 2n-punctured disk. We then describe two n-manifolds tildeS and tildeT in tildeC, and show that the Jones polynomial of L can be defined as an intersection pairing between tildeS and beta tildeT. Our construction is similar to one given by Lawrence, but more concrete.

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