Torus knots and the rational DAHA

Gorsky, Eugene; Oblomkov, Alexei; Rasmussen, Jacob; Shende, Vivek

doi:10.1215/00127094-2827126

Cited by 74 publications

(121 citation statements)

References 102 publications

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“…Remark 1.10. For torus knots, Theorem 1.9 was conjectured in [Gor12;ORS18;GORS14]. It was recently proved by the fourth author [Nak] based on the explicit computation of the Khovanov-Rozansky homology for torus knots [Mel].…”

Section: Introductionmentioning

confidence: 99%

Serre duality for Khovanov–Rozansky homology

Gorsky

Hogancamp

Mellit

et al. 2019

Sel. Math. New Ser.

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Section: Introductionmentioning

confidence: 99%

Serre duality for Khovanov–Rozansky homology

Gorsky

Hogancamp

Mellit

et al. 2019

Sel. Math. New Ser.

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“…We compute explicitly the sheaf corresponding to this braid. It is supported on the punctual Hilbert scheme: Hilb 0 1,n = fi Hilb The sheaf O Hilb 0 n (C 2 ) ⊗ det(B) k attracted a lot of attention in connection to combinatorics [24], representation theory [18] and knot theory [35,17]. In particular, in [35] it was conjectured that the global sections of this sheaf is a particular double-graded subspace of the space of Khovanov-Rozansky homology of the torus knot T n,1+nk .…”

mentioning

confidence: 99%

Knot homology and sheaves on the Hilbert scheme of points on the plane

Oblomkov

Rozansky

2018

Sel. Math. New Ser.

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175

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For each braid β ∈ Brn we construct a 2-periodic complex S β of quasi-coherent C * × C *equivariant sheaves on the non-commutative nested Hilbert scheme Hilb f ree 1,n . We show that the triply graded vector space of the hypercohomology H(S β ⊗ ∧ • (B)) with B being tautological vector bundle, is an isotopy invariant of the knot obtained by the closure of β. We also show that the support of cohomology of the complex S β is supported on the ordinary nested Hilbert scheme Hilb 1,n ⊂ Hilb f ree 1,n , that allows us to relate the triply graded knot homology to the sheaves on Hilb 1,n List of Symbols 81References 82

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“…Since there is a double product/sum in (2) and (3), it is natural to consider their refinement, where a second t-parameter is introduced, usually called q (for knots this means going from HOMFLY to super-and hyper-polynomials [69][70][71][72][73][74][75][76][77][78][79] …”

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confidence: 99%

On factorization of generalized Macdonald polynomials

Kononov

Морозов

2016

Eur. Phys. J. C

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A remarkable feature of Schur functions-the common eigenfunctions of cut-and-join operators from W ∞ -is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U q (SL N ) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorizationon a one-(rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.Generalized Macdonald polynomials (GMPs) [1-3] play a constantly increasing role in modern studies of the 6d version of AGT relations [31][32][33][34][35][36][37][38][39][40] and spectral dualities [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. At the same time they are relatively new special functions, far from being thoroughly understood and clearly described. They are deformations of the generalized Jack polynomials introduced in [59,60]. Even the simplest questions about them are yet unanswered. In this letter we address one of them-what happens to the hook formulas for classical, quantum, and Macdonald dimensions at the level of GMPs? We find that they survive, but only partly-on a onedimensional line in the space of time variables. Lifting to the entire two-dimensional topological locus remains to be found.We begin by recalling that the Schur functions χ R { p}, depend on representation (Young diagram) R and on infinitely many time variables p k (actually, a particular χ R depends a e-mail: morozov@itep.ru only on p k with k ≤ |R| = # boxes in R). They get nicely factorized on a peculiar two-dimensional topological locus,Coarm a and coleg l are the ordinary coordinates of the box in the diagram. To keep the notation consistent throughout the text, in (1) we call the relevant parameter t, not q, from the very beginning.It is often convenient to ignore the simple overall coefficient and substitute the product formulas like (2) by a polynomial expression for a plethystic logarithm ⎛

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Torus knots and the rational DAHA

Cited by 74 publications

References 102 publications

Serre duality for Khovanov–Rozansky homology

Serre duality for Khovanov–Rozansky homology

Knot homology and sheaves on the Hilbert scheme of points on the plane

On factorization of generalized Macdonald polynomials

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