We generalize the Khovanov-Rozansky cohomology for n=2 by means of a homogeneous potential that depends on two parameters, to obtain the universal Khovanov-Rozansky sl(2) link cohomology. This theory is equivalent to the universal foam sl(2) link cohomology, after tensoring both theories with appropriate rings.
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48 results for “Khovanov-Rozansky cohomology”
New operators in Khovanov-Rozansky homology exhibit symmetry.
problem Symmetry in Khovanov-Rozansky homology.
method Defined new commuting operators and proved satisfies hard Lefschetz property.
result Symmetry in Khovanov-Rozansky homology is confirmed.
We review Bennequin type inequalities established using various versions of the Khovanov-Rozansky cohomology. Then we give a new proof of a Bennequin type inequality established by the author, and derive new Bennequin type inequalities for knots using Gornik's version of the Khovanov-Rozansky cohomology, which generali…
Proves Serre duality in Khovanov-Rozansky homology.
problem Relating top and bottom Hochschild degrees in Khovanov-Rozansky homology.
method Using type A Soergel bimodules and full twist as a Serre functor.
result Categorifies Kálmán's theorem on Hochschild degrees.
We prove the quantum filtration on the Khovanov-Rozansky link cohomology H_p with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to H_p that is invariant under Reidemeister moves, whose E_1 term is isomorphic to the Khovanov-Roz…
Link homology compared with geometric link invariants using Bott-Samelson varieties.
problem Comparing different link homology theories with geometric link invariants.
method Using Khovanov-Rozansky homology and equivariant cohomology applied to Bott-Samelson varieties.
result Equivariant integral sl(n) link homology with specialized or universal potential.
New geometric model for knot homology using monodromic Hecke category.
problem Geometric description of Khovanov-Rozansky knot homology.
method Monodromic model based on Soergel bimodules and Hecke category.
result Geometric description of Khovanov-Rozansky knot homology.
We establish some inequalities about the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact which is sharper than the well known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants…
New stable homotopy types for knots and diagrams, aiding in their computation.
problem Computing Khovanov-Rozansky cohomology for knots and diagrams.
method Lifted simplified Khovanov-Rozansky complexes to stable homotopy types for open 2-braids.
result New stable homotopy types for knots and diagrams exhibit interesting behavior.
Simplified Khovanov-Rozansky operators for link invariants.
problem Complexity in Khovanov-Rozansky knot invariants.
method Substituted matrix factorization with simple operators and conjugations.
result Global reductions for simplifying link invariants.
We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved …
Research connects geometric structures to knot theory and algebraic combinatorics.
problem Understanding the mixed Hodge structure on cohomology of open positroid varieties.
method Relates mixed Hodge structure to Khovanov-Rozansky homology of associated links.
result Rational -Catalan numbers are derived from mixed Hodge polynomials of open positroid varieties.
For every positive integer we construct a bigraded homology theory for links, such that the corresponding invariant of the unknot is closely related to the U(n)-equivariant cohomology ring of ; our construction specializes to the Khovanov-Rozansky -homology. We are motivated by the "univers…
Torsion in Graph Homologymath.GT
Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is , we determine precisely those graphs whose cohomology contains torsion. For a la…
This study compares Landau-Ginzburg theory to Khovanov homology.
problem Comparing Landau-Ginzburg theory to Khovanov homology.
method Asymptotic analysis, exact WKB method, spectral networks method, null-webs.
result Landau-Ginzburg theory cohomology is equivalent to Khovanov homology.
It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in…
Remarks on some infinitesimal symmetries of Khovanov--Rozansky homologies in finite characteristicmath.GT
New proof of Khovanov-Rozansky homology base point independence in finite characteristic.
problem Base point independence of Khovanov-Rozansky homology in characteristic .
method New proof using infinitesimal symmetries of -homology.
result Proof of base point independence in characteristic .
Khovanov-Rozansky homology shows periodic links have group actions.
problem Understanding periodic links through homology.
method Showed Khovanov-Rozansky -homology has group actions on -periodic links.
result Proved an analog of periodicity criterion using -homology.
Simplified Khovanov-Rozansky calculus for bipartite knots.
problem Complexity in calculating superpolynomials for knots.
method Bipartite calculus generalizes Khovanov-Rozansky calculus for a restricted class of knots.
result Simplification of Khovanov-Rozansky polynomials for bipartite knots.
New 4-manifold invariants from Khovanov-Rozansky link homology.
problem Defining invariants for 4-manifolds.
method Using Khovanov-Rozansky gl(N) link homology, constructing skein modules from 4-categories with duals.
result Proof of the sweep-around property for well-defined link homologies in the 3-sphere.
2-Verma modules help categorify Khovanov-Rozansky homologies.
problem Categorification of Khovanov-Rozansky homologies.
method Using parabolic 2-Verma modules for .
result Construct higher representation theory of Khovanov-Rozansky homologies.
We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl_3-hom…
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
The paper connects knot homology with sheaf theory and proves symmetry properties.
problem Understanding the Khovanov-Rozansky homology of knots and links.
method Established an isomorphism between link homology and geometric homology, using sheaf theory.
result Proved the symmetry and computed Khovanov-Rozansky homology of torus links.
Spin Link Homologymath.QA
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.
New categorification method for infinite braids.
problem Categorifying highest-weight projectors for infinite braids.
method Limiting colored Khovanov-Rozansky homology of infinite braids.
result Partial isomorphism between HOMFLY-PT homology of braids and infinite torus knots.
Categorifies symmetric functions and computes invariants of tangles.
problem Computing and characterizing invariants of tangles.
method Using homotopy categories and symmetric monoidal categories.
result Existence of symmetric group actions and spectral sequences.
Functoriality proved for colored link invariants.
problem Link and tangle invariants functoriality proof.
method Functoriality proved for colored Khovanov-Rozansky invariants.
result Functoriality of colored link homologies proved.
New homology theory for annular knots and links.
problem No new problem introduced; extends existing Khovanov-Rozansky homology.
method Trace decategorification of enhanced sl(2) foams and categorified quantum gl(m) via classical skew Howe duality.
result Introduced sutured annular Khovanov-Rozansky homology with sl(n) action.
Computes a specific homology for a type of braid.
problem Calculating a homology for a particular class of braids.
method Uses Khovanov-Rozansky homology for Coxeter braids on 4 strands.
result Computed the triply graded Khovanov-Rozansky homology.
Study of 3-manifold homology using fivebrane compactifications.
problem Understanding homological invariants of 3-manifolds.
method Fivebrane compactifications and 3d/3d correspondence.
result Explicit calculations of 3-manifold homology invariants.
Torus link homologymath.GT
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.
problem Complexity reduction of Khovanov-Rozansky polynomial for bipartite links.
method Local reduction of matrix factorizations to planar cycles and simplification to vector spaces.
result KR polynomial for bipartite links simplifies to tensor products of vector spaces.
Analogous Morse moves for flow categories simplify flow data.
problem Simplifying flow data in Morse theory.
method Developed moves for framed flow categories analogous to Whitney trick and handle cancellation.
result Simplified flow categories can be computed by hand.
Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alter…
In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.
New homology for infinite multi-colored braids, completing previous work.
problem Categorification of highest-weight projectors for infinite braids.
method Defining limiting Khovanov-Rozansky homology for semi-infinite braids.
result Categorifies highest-weight projectors for a large class of braids.
We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.
Categorifies symmetric link invariants using foam technology.
problem Categorify symmetric link invariants using -webs.
method Finite dimensional categorification of symmetric evaluation of -webs using foam technology.
result Categorifies symmetric link invariants, providing a spectral sequence from Khovanov-Rozansky to symmetric.
The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homol…
New invariant restores symmetry in link homology and matches Hilbert scheme ideals.
problem Restoring missing symmetry in Khovanov-Rozansky homology.
method Defining a deformation of Khovanov-Rozansky homology with parameters .
result Matches ideals in Haiman's description of isospectral Hilbert scheme.
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All t…
We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of quantum invar…
Authors compute stable homology of torus knots using a new deformation technique.
problem Computing stable homology of torus knots.
method Link-splitting deformation (y-ification) of link homology.
result Explicit computation of -ified stable Khovanov--Rozansky homology of torus knots.
Homology of torus knotsmath.QA
Explicit formula found for torus knot homology.
problem Computing the homology of torus knots.
method Elias-Hogancamp method and combinatorics of toric braids.
result Explicit formula for triply graded Khovanov-Rozansky homology of torus knots.
We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fa…
New recursion computes knot homology linking Catalan sequences.
problem Computing Khovanov-Rozansky homology for specific knot families.
method Simple recursion to compute homology, interpreting results in terms of Catalan combinatorics.
result Agrees with predictions from Hilbert schemes and rational DAHA, proving conjectures.
New deformation of link homology for colored diagrams.
problem Understanding colored Khovanov-Rozansky homology.
method Introducing a multi-parameter deformation and extending to braids.
result Link splitting properties and invariants of colored Hopf links.