Orientations of Chord Diagrams and Khovanov Homology

The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satis...

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 construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a simple descri...

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov definition of Khovanov homology for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation ope...

This is an expository paper discussing various versions of Khovanov homology theories, interrelations between them, their properties, and their applications to other areas of knot theory and low-dimensional topology.

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. The goal of this note is to rewrite this construction in terms more friendly to topologists. A version of Khovanov homology for framed links is introduced. For...

This paper reviews and offers remarks upon Viro's definition of the Khovanov homology of the Kauffman bracket of unoriented framed tangles (Sec. 2). The review is based on a file of his talk. This definition contains an exposition of the relation between the $R$-matrix and the Kauffman bracket (Sec. 2.2).

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of signs when identifying variables in the $sl_2$-homology. Along with the behavior of the signs and local orientations in the $sl_2$-homology, both theories behave di...

Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and algebraically. In this paper we propose an alternative approach: we stay in the classical setup and fix the functoriality by simply adjusting the signs of the morphisms associated to the Reidemeister moves and the Morse moves.

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 crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.

In this paper, we study the Khovanov homology of an alternating virtual link $L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on the lines $j=2i-\sigma_{\xi}+2k-1$ for $0\leq k\leq g+1$ where $\sigma_{\xi^*}(L)+2g= \sigma_{\xi}(L)$ are the signatures of $L$ for a checkerboard coloring $\xi$ and its dual $\xi^*$. Of course, for classical links, the two signatures are equa...