arXiv research

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

Paper categorifies Vassiliev skein relation for Khovanov homology.

New modules derived from Khovanov homology for links.

Explains Khovanov homology and its applications.

Proves Khovanov homology functoriality and positivity for gl2 webs.

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…

We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial.

Khovanov homology distinguishes exotic 4-manifolds.

New method computes first Vassiliev derivative of Khovanov homology.

The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.

Skein lasagna module calculates 4-manifold invariants using handle decompositions.

New definition of skein lasagna module for specific 4-manifolds.

Khovanov homology ranks 2 for certain knots in a specific bundle.

New module constructs exotic surfaces in 4-manifolds.

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology.…

New method uses Khovanov homology to distinguish exotic 4-manifolds.

New spectral sequence connects Khovanov homology to real monopole Floer homology.

We compute the Khovanov lasagna module of S²×S², confirming a conjecture.

Let $Δ$ be a trivial knot in the three-sphere. For every finite cyclic group $G$ of odd order, we construct a $G$-equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to isotopy in $(S^{3},Δ)$. Another interpretation is given using the categorification of the …

Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to…

Study eight categorifications of colored Jones polynomial, verifying physics conjectures.

Extends quantum annular homology to infinite sets.

New mathematical tools for studying knots and links.

Researchers attempt to categorify biquandle brackets using Khovanov homology methods.

The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.

This paper defines a spectral sequence connecting knot homologies.

We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.

We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.

New invariant for 4-manifolds with framed links, stronger than existing invariants.

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructe…

Kirby color defined in Khovanov homology for 4D handlebodies.

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…

Quantum approach to volume computation from colored Jones polynomials.

The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant …

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphe…

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula fo…

X.S. Lin and O. Dasbach proved that the sum of the absolute value of the second and penultimate coefficients of the Jones polynomial of an alternating knot is equal to the twist number of the knot. In this paper we give a new proof of their result using Khovanov homology. The proof is by induction on the number of cros…

Expands Jones polynomial for Legendrian knots with categorification.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible ci…

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.

New deformation of link homology for colored diagrams.

Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link.…

Proves Khovanov homology has no torsion for bipartite circle graphs.

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the…

Functor decomposes Khovanov spectra for non-alternating diagrams.

Khovanov homology, an invariant of links in $\mathbb{R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and Sikora generalized this construction by defining a double graded homology theory…

New invariants derived from link homology for 4-manifolds.