DAHA and iterated torus knots
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.
Cite
@article{arxiv.1408.4348,
title = {DAHA and iterated torus knots},
author = {Ivan Cherednik and Ivan Danilenko},
journal= {arXiv preprint arXiv:1408.4348},
year = {2016}
}
Comments
v2: Editing, more material on the Betti numbers of Jacobian factors and non-algebraic knots was added, the relation to Jones polynomials was justified