DAHA and plane curve singularities

Cherednik, Ivan; Philipp, Ian

doi:10.2140/agt.2018.18.333

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DAHA and plane curve singularities

Ivan Cherednik and Ian Philipp

Algebraic & Geometric Topology 18 (2018) 333–385

DOI: 10.2140/agt.2018.18.333

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.

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

Volume 18, issue 1 (2018)