DAHA and iterated torus knots

TL;DR

This paper extends the DAHA-Jones polynomial theory to iterated torus knots, establishing new properties and conjecturing connections with algebraic knot invariants and singularity theory.

Contribution

It generalizes DAHA-Jones polynomials to arbitrary iterated torus knots and explores their properties and conjectural relations to algebraic geometry and knot invariants.

Findings

Polynomiality and duality of DAHA superpolynomials established.

Matches with classical algebraic knot theory and plane curve singularities.

Conjectured correspondence with Betti numbers of Jacobian factors.

Abstract

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities.