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DAHA and iterated torus
knots
Ivan Cherednik and Ivan Danilenko |
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Algebraic & Geometric Topology 16 (2016) 843–898
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Abstract |
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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 , are conjectured to provide the Betti numbers of the Jacobian factors (compactified Jacobians) of the corresponding singularities. |
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
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Mathematical Subject Classification 2010
Primary: 14H50, 17B45, 20C08, 57M25, 17B22
Secondary: 20F36, 33D52, 30F10, 55N10
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References |
Publication
Received: 19 December 2014
Revised: 5 June 2015
Accepted: 10 July 2015
Published: 26 April 2016
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Authors |
| Ivan Cherednik | |
| Department of Mathematics University of North Carolina Chapel Hill, NC 27599-3250 USA |
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| Ivan Danilenko | |
| MIPT 9 Institutskiy per. Dolgoprudny Moscow region 141700 Russia |
ITEP 25 Bolshaya Cheremushkinskaya Moscow 117218 Russia |
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