arXiv research

New algebraic framework for studying surfaces in 3-manifolds.

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.

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…

New module constructs exotic surfaces in 4-manifolds.

Explains Khovanov homology and its applications.

Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…

We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves …

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

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant und…

Spatial refinement of Bar-Natan homology constructed.

We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…

Unified invariant of knots derived from Verma modules.

Lower bounds on ribbon distance using Bar-Natan and α-Homology.

We show that the order of torsion homology classes in Bar-Natan deformation of Khovanov homology is a lower bound for the unknotting number. We give examples of knots that this is a better lower bound than |s(K)/2|, where s(K) is the Rasmussen s invariant defined by the Bar-Natan spectral sequence.

Kirby color defined in Khovanov homology for 4D handlebodies.

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

Develops a Gaussian model to compute the Alexander polynomial of knots.

Paper introduces Bar-Natan homology for special links in a modified space.

New link homology theories for $\mathbb{RP}^3$ yield distinct invariants.

We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to $\widetilde{\m…

Constructs a TQFT for 3-manifolds and extends it to 4-dimensional 2-handlebodies.

We show that over the binary field $\mathbb F_2$, the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch's analogous result for ordinary Khovanov homology, without the perturbation.

A spectral sequence is established, whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology, in…

New invariant distinguishes non-orientable surfaces.

New categories help understand knot algebra.

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the act…

Lecture notes on link homologies and knotted surfaces, focusing on 4D obstructions.

Formulae for Rasmussen invariant of satellite knots with wrapping number 2 proved.

A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ homology, in Khovanov'sclassification. Concordance invariants of…

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…

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 …

In this paper we prove the validity of a formula for computing the Alexander invariant which was originally conjectured by Bar-Natan and Dancso in [BND].

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

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.

Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.

Proves Khovanov homology conjecture for 3-stranded braids.

We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes …

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

Discrete Morse theory simplifies Khovanov homology calculations.

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characte…

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

We show how to use Bar-Natan's `divide and conquer' approach to computations to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

Algorithm simplifies Khovanov homology computations for 4-strand torus links.

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan, January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give, respectively,…

We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spiri…

Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. T…

Calculates knot $X$-torsion order using spectral sequences.