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A homological definition of the HOMFLY polynomial

Stephen Bigelow

Algebraic & Geometric Topology 7 (2007) 1409–1440

arXiv: arXiv:math/0608527

Abstract

We give a new definition of the knot invariant associated to the Lie algebra suN+1. Knowing these for all N is equivalent to knowing the HOMFLY polynomial. Our definition requires that the knot or link be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two immersed manifolds in a configuration space of points in a disk. This generalizes previous work on the Jones polynomial, which is the case N = 1.

Keywords
HOMFLY polynomial, braid group, plat closure, bridge position, configuration space
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 20F36
References
Publication
Received: 23 August 2006
Revised: 14 September 2007
Accepted: 14 September 2007
Published: 15 October 2007
Authors
Stephen Bigelow
Department of Mathematics
University of California at Santa Barbara
California 93106
USA
http://www.math.ucsb.edu/~bigelow/