Volume 18, issue 1 (2018)

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DAHA and plane curve singularities

Ivan Cherednik and Ian Philipp

Algebraic & Geometric Topology 18 (2018) 333–385
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 and Mazin for their constant term. The paper mainly focuses on nontorus algebraic knots. A connection with the conjecture due to Oblomkov, Rasmussen and 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.

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Keywords
Hecke algebra, Jones polynomial, HOMFLYPT polynomial, Khovanov-Rozansky homology, algebraic knot, Macdonald polynomial, plane curve singularity, compactified Jacobian, Puiseux expansion, orbital integral
Mathematical Subject Classification 2010
Primary: 14H50, 17B45, 20C08, 20F36, 33D52, 57M25
Secondary: 17B22, 22E50, 22E57, 30F10, 33D80
Supplementary material

Appendices A and B

References
Publication
Received: 28 October 2016
Revised: 15 May 2017
Accepted: 12 June 2017
Published: 10 January 2018
Authors
Ivan Cherednik
Mathematics Department
University of North Carolina
Chapel Hill, NC
United States
Ian Philipp
Mathematics Department
University of North Carolina
Chapel Hill, NC
United States