Algebraic & Geometric Topology

DAHA and plane curve singularities

Ivan Cherednik and Ian Philipp

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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.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 1 (2018), 333-385.

Dates
Received: 28 October 2016
Revised: 15 May 2017
Accepted: 12 June 2017
First available in Project Euclid: 1 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1517454219

Digital Object Identifier
doi:10.2140/agt.2018.18.333

Mathematical Reviews number (MathSciNet)
MR3748246

Zentralblatt MATH identifier
06828007

Subjects
Primary: 14H50: Plane and space curves 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] 20C08: Hecke algebras and their representations 20F36: Braid groups; Artin groups 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 17B22: Root systems 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 22E57: Geometric Langlands program: representation-theoretic aspects [See also 14D24] 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 33D80: Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics

Keywords
Hecke algebra Jones polynomial HOMFLYPT polynomial Khovanov-Rozansky homology algebraic knot Macdonald polynomial plane curve singularity compactified Jacobian Puiseux expansion orbital integral

Citation

Cherednik, Ivan; Philipp, Ian. DAHA and plane curve singularities. Algebr. Geom. Topol. 18 (2018), no. 1, 333--385. doi:10.2140/agt.2018.18.333. https://projecteuclid.org/euclid.agt/1517454219


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Supplemental materials

  • Appendices A and B.