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2016 DAHA and iterated torus knots
Ivan Cherednik, Ivan Danilenko
Algebr. Geom. Topol. 16(2): 843-898 (2016). DOI: 10.2140/agt.2016.16.843

Abstract

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which includes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov–Rozansky polynomials in the case of nonnegative 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 (compactified Jacobians) of the corresponding singularities.

Citation

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Ivan Cherednik. Ivan Danilenko. "DAHA and iterated torus knots." Algebr. Geom. Topol. 16 (2) 843 - 898, 2016. https://doi.org/10.2140/agt.2016.16.843

Information

Received: 19 December 2014; Revised: 5 June 2015; Accepted: 10 July 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1375.14099
MathSciNet: MR3493410
Digital Object Identifier: 10.2140/agt.2016.16.843

Subjects:
Primary: 14H50, 17B22, 17B45, 20C08, 57M25
Secondary: 20F36, 30F10, 33D52, 55N10

Rights: Copyright © 2016 Mathematical Sciences Publishers

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