The Khovanov complex for virtual links

One of the most outstanding achievements of the modern knot theory is Khovanov's categorification of Jones polynomials. In the present paper, we construct the homology theory for virtual knots. An important obstruction to this theory (unlike the case of classical knots) is the nonorientability of “atoms”; an atom is a two-dimensional combinatorial object closely related with virtual link diagrams. The problem is solved directly for the field $\mathbb Z_{2}$, and also by using some geometrical constructions applied to atoms. We discuss a generalization proposed by Khovanov; he modifies the initial homology theory by using the Frobenius extension. We construct analogues of these theories for virtual knots, both algebraically and geometrically (by using atoms).