The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Abstract

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.

with an Appendix by Eugene Gorsky

References in corpus (8)

• Knot Homology from Refined Chern-Simons Theory
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Toric braids and $(m,n)$-parking functions
• Torus knots and the rational DAHA
• Finite dimensional representations of rational Cherednik algebras
• A support theorem for Hilbert schemes of planar curves
• Khovanov-Rozansky homology and higher Catalan sequences
• Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial

Cited by in corpus (23)

• Legendrian knots and constructible sheaves
• Homology of torus knots
• DAHA and iterated torus knots
• Categorified Young symmetrizers and stable homology of torus links
• Cabling procedure for the colored HOMFLY polynomials
• Rational links and DT invariants of quivers
• Challenges of beta-deformation
• Tangle addition and the knots-quivers correspondence
• Torus link homology
• Critical behavior in topological ensembles
• The action of full twist on the superpolynomial for torus knots
• The Harish-Chandra isomorphism for quantum GL_2
• A support theorem for Hilbert schemes of planar curves, II
• Modules over plane curve singularities in any ranks and DAHA
• Dualizable link homology
• Tautological classes and symmetry in Khovanov-Rozansky homology
• Affine Springer Fibers and Generalized Haiman Ideals
• From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
• The Hilb-vs-Quot Conjecture
• Refined Hilbert schemes, E-polynomials, and the number of generators of finite colength ideals in the plane
• Algebra and geometry of link homology
• Flops and Hilbert schemes of space curve singularities
• Superpolynomials of algebraic links