Khovanov homology for virtual knots with arbitrary coefficients

The Khovanov homology theory over an arbitrary coefficient ring is extended to the case of virtual knots. We introduce a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovanov complex in the classical case. Unlike Khovanov's original construction, our definition of the complex does not use any additional prescription of signs to the edges of a cube. Moreover, our method enables us to construct a Khovanov homology theory for 'twisted virtual knots' in the sense of Bourgoin and Viro (including knots in three-dimensional projective space). We generalize a number of results of Khovanov homology theory (the Wehrli complex, minimality problems, Frobenius extensions) to virtual knots with non-orientable atoms