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DAHA and iterated torus knots

Ivan Cherednik and Ivan Danilenko

Algebraic & Geometric Topology 16 (2016) 843–898

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.

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
Mathematical Subject Classification 2010
Primary: 14H50, 17B45, 20C08, 57M25, 17B22
Secondary: 20F36, 33D52, 30F10, 55N10
Received: 19 December 2014
Revised: 5 June 2015
Accepted: 10 July 2015
Published: 26 April 2016
Ivan Cherednik
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599-3250
Ivan Danilenko
9 Institutskiy per.
Moscow region
25 Bolshaya Cheremushkinskaya