  We conjecture an expression for the dimensions of the Khovanov-Rozansky
HOMFLY homology groups of the link of a plane curve singularity in terms of the
weight polynomials of Hilbert schemes of points scheme-theoretically supported
on the singularity. The conjecture specializes to our previous conjecture
relating the HOMFLY polynomial to the Euler numbers of the same spaces upon
setting t = -1. By generalizing results of Piontkowski on the structure of
compactified Jacobians to the case of Hilbert schemes of points, we give an
explicit prediction of the HOMFLY homology of a (k, n) torus knot as a certain
sum over diagrams.
  The Hilbert scheme series corresponding to the summand of the HOMFLY homology
with minimal "a" grading can be recovered from the perverse filtration on the
cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this
space furnishes the unique finite dimensional simple representation of the
rational spherical Cherednik algebra with central character k/n. Up to a
conjectural identification of the perverse filtration with a previously
introduced filtration, the work of Haiman and Gordon and Stafford gives
formulas for the Hilbert scheme series when k = mn + 1.
