A homological definition of the HOMFLY polynomial (Q2464799)

In a previous paper [Geom. Topol. Monogr. 4, 29--41 (2002; Zbl 1035.57004)], the author gave a definition of the Jones polynomial as a homological intersection pairing between a certain pair of manifolds in a configuration space. The purpose of the paper under review is to give a similar definition of the HOMFLY polynomial. More precisely, a specialization of the HOMFLY polynomial, which is called the invariant of type \(A_N\), is considered. The invariant \(P\) of type \(A_N\) satisfies the skein relation \[ q^{(N+1)/2}P(K_-)-q^{-(N+1)/2}P(K_+)=(q^{1/2}-q^{-1/2})P(K_0), \] where \(K_+\), \(K_-\) and \(K_0\) are the usual skein triples, and takes the value one for the unknot. The case where \(N=1\) corresponds to the Jones polynomial, and the HOMFLY polynomial is reconstructed from the values for all \(N\). The definition requires that the knot or link is represented as the plat closure of a braid. A certain configuration space of points in a disk is defined, and a homological intersection pairing between two immersed manifolds there gives the invariant.