ABSTRACT
This chapter constructs the Khovanov homology for virtual knots. The Khovanov homology possesses important properties coming from algebraic topology: the functoriality. One of the most natural problems in the theory of virtual knots is the problem of generalisation of the Khovanov complex for virtual knots. The chapter also constructs a chain complex for a virtual diagram with the homology being invariant under the generalised Reidemeister moves. It aims to change the basis of the Frobenius algebra representing the Khovanov homology of the unknot as we pass from one crossing of the knot diagram to another. The chapter explores to replace the usual tensor product by the exterior product of the corresponding graded spaces. It shows how one can establish the minimality of link diagrams by using the Khovanov complex. Different minimality theorem will be formulated which are based on the Jones polynomial as well as the Khovanov complex.
