Corpus ID: 119612787

# Hilbert schemes and $y$-ification of Khovanov-Rozansky homology

@article{Gorsky2017HilbertSA,
title={Hilbert schemes and \$y\$-ification of Khovanov-Rozansky homology},
author={Eugene Gorsky and Matthew Hogancamp},
journal={arXiv: Geometric Topology},
year={2017}
}
• Published 11 December 2017
• Mathematics
• arXiv: Geometric Topology
Author(s): Gorsky, Eugene; Hogancamp, Matthew | Abstract: We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $\mathbb{Q}[x_c,y_c]_{c\in \pi_0(L)}$. We conjecture that this invariant restores the missing $Q… Expand Soergel bimodules and matrix factorizations. • Mathematics, Physics • 2020 We establish an isomorphism between the Khovanov-Rozansky triply graded link homology and the geometric triply graded homology due to the authors. Hence we provide an interpretation of theExpand Dualizable link homology • Mathematics • 2019 We modify our previous construction of link homology in order to include a natural duality functor$\mathfrak{F}$. To a link$L$we associate a triply-graded module$HXY(L)$over the gradedExpand Unramified affine Springer fibers and isospectral Hilbert schemes For any connected reductive group$G$over$\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibersExpand From the Hecke Category to the Unipotent Locus Let W be the Weyl group of a split semisimple group G. Its Hecke category HW can be built from pure perverse sheaves on the double flag variety of G. By developing a formalism of generalizedExpand Categorical Chern character and braid groups. • Mathematics • 2018 To a braid$\beta\in Br_n$we associate a complex of sheaves$S_\beta$on$Hilb_n(C^2)$such that the previously defined triply graded link homology of the closure$L(\beta)$is isomorphic to theExpand Curved Rickard complexes and link homologies • Mathematics • 2019 Abstract Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which weExpand Ext-enhanced monoidal Koszul duality for$\mathrm{GL}_2$. • Mathematics • 2019 The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recentExpand Notes on Matrix Factorizations and Knot Homology These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triplyExpand A G ] 2 3 A ug 2 02 1 Algebra and geometry of link homology Lecture notes from the IHES 2021 Summer School 3 Khovanov-Rozansky homology: definitions and computations 6 3.1 Soergel bimodules and Rouquier complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Khovanov-Rozansky homology . . .Expand On monoidal Koszul duality for the Hecke category We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche,Expand #### References SHOWING 1-10 OF 24 REFERENCES Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology • Mathematics • 2016 Author(s): Gorsky, Eugene; Neguţ, Andrei; Rasmussen, Jacob | Abstract: We construct a categorification of the maximal commutative subalgebra of the type$A$Hecke algebra. Specifically, we propose aExpand Khovanov homology and knot Floer homology for pointed links • Mathematics • 2015 A well-known conjecture states that for any$l$-component link$L$in$S^3$, the rank of the knot Floer homology of$L$(over any field) is less than or equal to$2^{l-1}$times the rank of theExpand Khovanov-Rozansky homology and higher Catalan sequences We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the$(n,nm\pm 1)$and$(n,nm)$torus links forExpand The Superpolynomial for Knot Homologies • Mathematics, Physics • Exp. Math. • 2006 A framework for unifying the sl(N) Khovanov– Rozansky homology with the knot Floer homology is proposed, and a rich formal structure is proposed that is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. Expand Hilbert schemes, polygraphs and the Macdonald positivity conjecture We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n isExpand Torus link homology and the nabla operator • A. T. Wilson • Mathematics, Computer Science • J. Comb. Theory, Ser. A • 2018 A combinatorial formula for the homologies of all links considered by Elias and Hogancamp and conjecture a direct relationship between the$(n,n)$torus link case of the formula and the symmetric function$\nabla p_{1^n}\$. Expand
Knot homology and sheaves on the Hilbert scheme of points on the plane
• Mathematics
• 2016
For each braid $$\beta \in \mathfrak {Br}_n$$β∈Brn we construct a 2-periodic complex $$\mathbb {S}_\beta$$Sβ of quasi-coherent $$\mathbb {C}^*\times \mathbb {C}^*$$C∗×C∗-equivariant sheaves on theExpand
Macdonald Polynomials and Geometry