Hilbert schemes of singular plane curves and HOMFLY homology of their links
Seminar Room 1, Newton Institute
Intersecting a plane curve with the boundary of a small ball around one of its singularities yields a link in the 3-sphere. To any link may be attached a triply graded vector space, the HOMFLY homology. Taking its Euler characteristic with respect to a certain grading gives the HOMFLY polynomial, which in turn specializes variously to the Alexander polynomial, the Jones polynomial, and the other SU(n) knot polynomials.
We will present a conjecture recovering this invariant from moduli spaces attached to the singular curve. Specifically, we form the Hilbert schemes of points of the curve, and certain incidence varieties inside products of Hilbert schemes. Up to certain shifts of grading, we conjecture that the HOMFLY homology of the link of the singularity is the direct sum of the homologies of these spaces.
This talk presents joint work with J. Rasmussen and A. Oblomkov.