A survey of Bar-Natan skein theory

Bar-Natan skein modules have been introduced by Bar-Natan in his approach to Khovanov homology. In his case these are quotients of abelian groups generated by surfaces, with components possibly carrying dots, by a set of certain surgery induced relations. These modules have been generalized by Asaeda and Frohman to a setting of dotted surfaces embedded in compact 3-manifolds. I will discuss a general definition of Bar-Natan modules for surfaces embedded in compact 3-manifolds, with the components colored by elements of an arbitrary commutative Frobenius algebra (with the dot case corresponding to x in Z[x]/(x^2)). This construction defines in fact a full extension of the 2-dimensional TQFT corresponding to the Frobenius algebra. Then I will show that each commutative Frobenius algebra defines a natural category of algebras. Moreover, for each compact 3-manifold with a closed 1-manifold in its boundary, there is a compression body category defined by Casson and Gordon. We will define a functor from the compression body category into this category of algebras, such that the colimit of this functor is the Bar-Natan module. This new type of categorification leads to a good understanding of how the topology of the 3-manifold is reflected in the modules and to presentations in terms of incompressible surfaces (in the dotted case already observed by Asaeda and Frohman). There is a bicategory version of this result, an orientable and an oriented version, and functors relating these structures. If time permits I will also discuss the definition and some results and questions for a skein module combining Bar-Natan skein modules with the Kauffman bracket skein module.
Zoom 875 9016 0151, the usual password (ask V. M. Nezhinskij: nezhin@pdmi.ras.ru).