DAHA and plane curve singularities
Abstract
We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik-Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky-Mazin for their constant term. The paper mainly focuses on non-torus algebraic knots. A connection with the conjecture due to Oblomkov-Rasmussen-Shende is possible, but our approach is different. A motivic version of our conjecture is related to p-adic orbital A-type integrals for anisotropic centralizers.
Cite
@article{arxiv.1605.00978,
title = {DAHA and plane curve singularities},
author = {Ivan Cherednik and Ian Philipp},
journal= {arXiv preprint arXiv:1605.00978},
year = {2018}
}
Comments
v2: extensive editing, added refs, an extension toward orbital integrals; v3: some editing, confirmations for non-affine cells; v4: essentially equivalent to the AGT version