Khovanov homology for 3-manifolds.

Dror Bar-Natan's approach to understanding the Khovanov homology leads
to local relations on the level of surfaces. Using these relations we
define a graded module associated to a three-manifold built out of
surfaces in the manifold,
modulo Bar-Natan's relations.  For hyperbolic manifolds the graded
dimension of the module is a rational function. We go on to define
Khovanov homology for a link in the boundary of a three-manifold, the
graded Euler characteristic of the theory for the unknot is the
rational function defined above. (This work was joint with Marta
Asaeda.)