Khovanov homology invariant under Conway mutation.
problem Invariance of Khovanov homology under specific transformations.
method Strong geography restrictions and homological mirror symmetry.
result Classification of components of a Khovanov multicurve invariant.
A new skein relation for Khovanov homology categorifies the θ-invariant.
problem Categorify the θ-invariant using Khovanov homology.
method Developed a skein relation for Khovanov homology using spectral sequences.
result Categorified the θ-invariant using Khovanov homology.
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
New concordance invariant from spectral sequence on Khovanov homology.
problem Concordance invariants from spectral sequences.
method Constructing from E(−1) spectral sequence on Khovanov homology. result Provides a bound on the nonorientable slice genus.
We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We a…
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
New mathematical tools for studying knots and links.
problem Understanding knot and link diagrams using topological invariants.
method Introducing Khovanov Laplacian and Khovanov Dirac to study diagrams.
result The harmonic spectrum retains Khovanov homology invariants, while non-harmonic spectra reveal additional information.
Khovanov homology invariant proved for links in RP3.
problem Proving Khovanov homology invariant for links in RP3. method Established functoriality of Khovanov homology for link cobordisms in IimesRP3. result Khovanov homology is an invariant of links in unparametrized RP3. Paper categorifies Vassiliev skein relation for Khovanov homology.
problem Clarifying the relation between Vassiliev invariants and Khovanov homology.
method Developed a categorified version of Vassiliev skein relation on Khovanov homology.
result Khovanov homology's genus-one operation leads to a crossing change, enabling invariance under Reidemeister moves and extending to singular links.
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…
We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka's instanton invariants J♯ and I♯ for spatial webs by two spectral sequences. As an application of the spectral seq…
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and the linking numbers of its com…
In this article, we prove the conjecture of Bar-Natan, Garoufalidis, and Khovanov's on the support of the Khovanov's invariants for alternating knots.
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surf…
Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …
Enhanced Khovanov TQFT using basepoints for nonorientable surfaces.
problem Invariance of nonorientable surfaces in Khovanov homology.
method Basepoint actions in Khovanov homology over the field with two elements.
result Establishes invariance under connected sums with specific surfaces.
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…
Proof of isotopy invariance in Khovanov link homology.
problem Isotopy invariance of Khovanov link homology.
method Explicit description of retractions and chain homotopies.
result Isotopy invariance of Khovanov link homology proved.
Defines an odd analog of Plamenevskaya's invariant for transverse links.
problem No specific problem stated; focuses on extending an invariant.
method Defines and analyzes an odd analog invariant in Khovanov homology.
result The odd analog invariant is an invariant of transverse links with similar properties.
Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.
problem Detecting and obstructing sliceness of knots using surface invariants.
method Defining an invariant from Khovanov homology of boundary links.
result The invariant can obstruct sliceness of knots and detect slices for specific knots.
O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these…
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.
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
New bounds for knot distances using Khovanov homology.
problem Calculating precise distances between knots.
method Using Khovanov homology to refine existing bounds.
result Improved bounds for Gordian distances of knots.
Proposes a method to compute the second Steenrod square for odd Khovanov homology.
problem Computing the second Steenrod square for odd Khovanov homology.
method Proposes a new method to compute the second Steenrod square, showing it to be a link invariant.
result Shows the proposed method gives a refinement of the Rasmussen s-invariant with Z/2Z coefficients. Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in Z2 generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras…
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtrat…
Researchers link tangle invariants for Khovanov and knot Floer homologies.
problem Relating tangle invariants for Khovanov and knot Floer homologies.
method Constructing algebraic DA bimodules for tangles and open braids, showing homotopy equivalence to Ozsvath-Szabo bimodules.
result Homotopy equivalence of DA bimodules for tangles and knot Floer homology.
Khovanov homology extended to 3-manifolds, linking tangles.
problem Extending Khovanov homology to 3-manifolds.
method Cutting 3-manifolds into intervals, constructing type D and A structures, gluing back.
result Recovering Khovanov homology of links in 3-manifolds.
We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There i…
Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
We prove that Khovanov homology and Lee homology with coefficients in F2 are invariant under component-preserving link mutations.
Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are…
Explains Khovanov homology and its applications.
problem Understanding Khovanov homology and its applications.
method Expository lecture notes covering Jones polynomial, Khovanov homology, cobordism category, spectral sequences, and skein lasagna modules.
result Explains the Jones polynomial, Khovanov homology, and their applications.
Researchers compute Khovanov polynomials for satellite knots.
problem Computing Khovanov polynomials for satellite knots.
method Explicit computation using a computer program for two families of satellite knots.
result Khovanov polynomials can be expressed as a linear combination of pattern and companion invariants, with a jump at a critical point.
New invariant for 4-manifolds with framed links, stronger than existing invariants.
problem Distinguishing 4-manifolds with framed links.
method Introducing a new invariant called KLS lasagna homotopy type.
result The new invariant is stronger than existing invariants.
As Oleg Viro describes in his paper, the most fundamental property of the Khovanov homology group is their invariance under Reidemeister moves. Viro constructes Khovanov complex and homology consisting of Jordan curves with sign and also gives a proof for the only case of first Reidemeister move by using his definition…
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…
We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
Khovanov multicurves are restricted to linear components.
problem Khovanov multicurves are subject to strong geography restrictions.
method Applying ideas from homological mirror symmetry, we show that Khovanov multicurves are linear.
result Every component of the invariant is linear, homotopic to a straight line.
New homomorphism from Khovanov homology gives slice genus bounds.
problem Understanding slice genus of knots.
method A 1-parameter family of concordance homomorphisms from Khovanov homology.
result Can prove linear independence of certain knot families.
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.