Nilpotent slices, Hilbert schemes, and the Jones polynomial

Abstract

Seidel and Smith [33] have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety $Y.$ This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra ${\mathfrak{sl}}_{2m}.$ We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting $Y$ as an open subset of the Hilbert scheme of a Milnor fiber