Translator Disclaimer
2018 The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
Alexei Oblomkov, Jacob Rasmussen, Vivek Shende
Geom. Topol. 22(2): 645-691 (2018). DOI: 10.2140/gt.2018.22.645

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 (2012) 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 kn. 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.

Citation

Download Citation

Alexei Oblomkov. Jacob Rasmussen. Vivek Shende. "The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link." Geom. Topol. 22 (2) 645 - 691, 2018. https://doi.org/10.2140/gt.2018.22.645

Information

Received: 14 September 2012; Accepted: 20 April 2017; Published: 2018
First available in Project Euclid: 1 February 2018

zbMATH: 06828597
MathSciNet: MR3748677
Digital Object Identifier: 10.2140/gt.2018.22.645

Subjects:
Primary: 14H20, 14N35
Secondary: 57M27

Rights: Copyright © 2018 Mathematical Sciences Publishers

JOURNAL ARTICLE
47 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.22 • No. 2 • 2018
MSP
Back to Top