  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 description of Khovanov long exact sequence of homology.
Khovanov homology, over the field Q, is a categorification of the Jones
polynomial (i.e. one represents the Jones polynomial as the generating function
of Euler characteristics). However, for integral coefficients Khovanov homology
almost always has torsion. The first part of the chapter is devoted to the
construction of torsion in Khovanov homology. In the second part we analyze the
thickness of Khovanov homology and reduced Khovanov homology. In the last part
we generalize Khovanov homology to links in $I$-bundles over surfaces and
demonstrate that for oriented surfaces we categorify the element of the
Kauffman bracket skein module represented by the considered link.
