A homological definition of the HOMFLY polynomial

Bigelow, Stephen

doi:10.2140/agt.2007.7.1409

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A homological definition of the HOMFLY polynomial

Stephen Bigelow

Algebraic & Geometric Topology 7 (2007) 1409–1440

DOI: 10.2140/agt.2007.7.1409

arXiv: arXiv:math/0608527

Abstract

We give a new definition of the knot invariant associated to the Lie algebra ${s u}_{N + 1}$ . 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/

Volume 7, issue 3 (2007)