Graphs on surfaces and Khovanov homology

The Chromatic Polynomial of Fatgraphs and Its Categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homolog...

When the Theories Meet: Khovanov Homology as Hochschild Homology of Links

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2, n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, sl(n), homology and their deforma...

Surfaces with pulleys and Khovanov homology

Link Homology and Unoriented Topological Quantum Field Theory

We investigate Khovanov homology of stable equivalence classes of link diagrams on oriented surfaces. We apply Bar-Natan’s geometric formalism to this setting and using unoriented 1+1-dimensional topological quantum field theories we define new link homology theories for stable equivalences classes.

An Invariant of Link Cobordisms from Khovanov’s Homology Theory

i,j(−1) q rkH (D) is the Jones polynomial of L. The groups H (D) were defined as homology groups of a bigraded chain complex. Khovanov proved that the homology groups considered up to isomorphism depend only on the link L. In fact, he showed that any Reidemeister move induces a chain equivalence (of bidegree (0, 0)) between the chain complexes of the two diagrams involved. Khovanov acknowledged...