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Link splitting deformation of colored Khovanov--Rozansky homology
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Abstract
We introduce a multi-parameter deformation of the triply-graded Khovanov--Rozansky homology of links colored by one-column Young diagrams, generalizing the "y-ified" link homology of Gorsky--Hogancamp and work of Cautis--Lauda--Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2-category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from arXiv:2107.08117 and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply-graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov--Rozansky homology and Hilbert schemes.
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We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.
We prove that the bigraded colored Khovanov-Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials on these dg algebras and conjecture that their homology matches that of the projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.
We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links with Hilbert schemes.
We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Sto\v{s}i\'c about the growth of colored HOMFLY-PT homologies.
We show that canonical bases in and the Schur
algebra are compatible; in fact we extend this result to p-canonical bases.
This follows immediately from a fullness result from a functor categorifying
this map. In order to prove this result, we also explain the connections
between categorifications of the Schur algebra which arise from parity sheaves
on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's
"flag category," which are of some independent interest.
We reconsider the su(3) link homology theory defined by Khovanov in
math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With
some slight modifications, we describe the theory as a map from the planar
algebra of tangles to a planar algebra of (complexes of) `cobordisms with
seams' (actually, a `canopolis'), making it local in the sense of Bar-Natan's
local su(2) theory of math.GT/0410495.
We show that this `seamed cobordism canopolis' decategorifies to give
precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined
in q-alg/9712003. We conjecture an answer to an even more interesting question
about the decategorification of the Karoubi envelope of our cobordism theory.
Finally, we describe how the theory is actually completely computable, and
give a detailed calculation of the su(3) homology of the (2,n) torus knots.
We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl.3/ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl.2/ invariant (or Lee's modification of it) corresponds to a D 0 (or a D 1/: In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl.2/ theory.
The Murphy operators in the Hecke algebra H n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H n . In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, [Formula: see text].
Symmetric functions of the Murphy operators are also central in H n . I define geometrically a homomorphism from [Formula: see text] to the centre of each algebra H n , and find an element in [Formula: see text], independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of [Formula: see text] in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.
We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have `seams' separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter and Saito's movie moves (MR1238875, MR1445361) always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a `local' result about tangles. Along the way, we reproduce Jacobsson's sign table (math.GT/0206303) for the original `unoriented theory', with a few disagreements. Comment: 91 pages. Added David Clark as co-author. Further detail on variations of third Reidemeister moves, to allow treatment of previously missing cases of movie move six. See changelog section for more detail
We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan - Lusztig theory, and which describe a direct summand of the category of Harish - Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.
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 we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y -ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).
We show that the triply graded Khovanov–Rozansky homology of the torus link Tn,k stabilizes as k →∞. We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky, Oblomkov, Rasmussen and Shende. To accomplish this, we construct complexes Pn of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that Pn is a stable limit of Rouquier complexes. A certain derived endomorphism ring of Pn computes the aforementioned stable homology of torus links.
We study the properties of the extended graphical calculus for categorified quantum sl(n). The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and arbitrary colors -- the results that were anounced in [M. Stosic: Indecomposable objects and Lusztig's canonical basis, Math. Res. Lett. 22, no. 1 (2015), 245-278].
We introduce an sl(n) homology theory for knots and links in the thickened
annulus. To do so, we first give a fresh perspective on sutured annular
Khovanov homology, showing that its definition follows naturally from trace
decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via
classical skew Howe duality. This framework then extends to give our annular
sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky
homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of
an annular link carries an action of the Lie algebra sl(n), which in the n=2
case recovers a result of Grigsby-Licata-Wehrli.
We give a purely combinatorial construction of colored link
homology. The invariant takes values in a 2-category where 2-morphisms are
given by foams, singular cobordisms between webs; applying a
(TQFT-like) representable functor recovers (colored) Khovanov-Rozansky
homology. Novel features of the theory include the introduction of `enhanced'
foam facets which fix sign issues associated with the original matrix
factorization formulation and the use of skew Howe duality to show that
(enhanced) closed foams can be evaluated in a completely combinatorial manner.
The latter answers a question posed in math.GT/0708.2228.
We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer and Mrowka and Hedden and Ni to show that Khovanov homology detects the unlink.
The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z \mathbb {Z} -graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category O \mathcal {O} that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O \mathcal {O} again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.
We describe categorifications of sl 2 and braid groups. In a first part, we give a survey of the case of sl 2 (joint work with Joseph Chuang [ChRou]) and explain how it leads to the construction of derived equiva-lences. The second part points out the existence of "higher symmetries" in the examples of braid group actions on triangulated categories.
Khovanov has given a construction of the Khovanov-Rozansky link invariants
(categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid
groups. We give a direct proof that his construction does give link invariants.
We show more generally that, for any finite Coxeter group, his construction
provides a Markov "2-trace", and we actually show that the invariant takes
value in suitable derived categories. This makes more precise a result of
Trafim Lasy who has shown that, after taking the class in K_0, this provides a
Markov trace.
We define and study categories of singular Soergel bimodules, which are
certain natural generalisations of Soergel bimodules. Indecomposable singular
Soergel bimodules are classified, and we conclude that the split Grothendieck
group of the 2-category of singular Soergel bimodules is isomorphic to the
Schur algebroid. Soergel's conjecture on the characters of indecomposable
Soergel bimodules in characteristic zero is shown to imply a similar conjecture
for the characters of singular Soergel bimodules.
Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators. Comment: 40 pages, many figures
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 is normal, Cohen-Macaulay, and Gorenstein, and hence flat over H_n. We derive two important consequences. (1) We prove the strong form of the "n! conjecture" of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K_{lambda,mu}(q,t). This establishes the Macdonald positivity conjecture, that K_{lambda,mu}(q,t) is always a polynomial with non-negative integer coefficients. (2) We show that the Hilbert scheme H_n is isomorphic to the Hilbert scheme of orbits C^2n//S_n, in such a way that X_n is identified with the universal family over C^2n//S_n.
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