A homological definition of the HOMFLY polynomial

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 $N=1$.