Volume 7, issue 3 (2007)

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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 ${\mathfrak{s}\mathfrak{u}}_{N+1}$. 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