Algebraic & Geometric Topology

DAHA and iterated torus knots

Ivan Cherednik and Ivan Danilenko

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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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 843-898.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841155

Digital Object Identifier
doi:10.2140/agt.2016.16.843

Mathematical Reviews number (MathSciNet)
MR3493410

Zentralblatt MATH identifier
1375.14099

Subjects
Primary: 14H50: Plane and space curves 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] 20C08: Hecke algebras and their representations 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 17B22: Root systems
Secondary: 20F36: Braid groups; Artin groups 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 55N10: Singular theory

Keywords
double affine Hecke algebra Jones polynomials HOMFLY-PT polynomial Khovanov-Rozansky homology iterated torus knot cabling Macdonald polynomial plane curve singularity generalized Jacobian Betti numbers Puiseux expansion

Citation

Cherednik, Ivan; Danilenko, Ivan. DAHA and iterated torus knots. Algebr. Geom. Topol. 16 (2016), no. 2, 843--898. doi:10.2140/agt.2016.16.843. https://projecteuclid.org/euclid.agt/1510841155


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