## Algebraic & Geometric Topology

### 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 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
Revised: 15 May 2017
Accepted: 12 June 2017
First available in Project Euclid: 1 February 2018

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

#### 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

#### Supplemental materials

• Appendices A and B.