DAHA and plane curve singularities

TL;DR

This paper proposes a geometric conjecture linking uncolored DAHA superpolynomials of algebraic knots to flagged Jacobian factors of plane curve singularities, extending previous conjectures and focusing on non-torus knots.

Contribution

It introduces a new geometric framework for describing DAHA superpolynomials using flagged Jacobian factors, generalizing several existing conjectures.

Findings

Conjectural description of superpolynomials via Jacobian factors

Extension of Gorsky's combinatorial interpretation to algebraic knots

Potential connections to p-adic orbital integrals

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.