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We construct explicitly the Khovanov homology theory for virtual links with arbitrary coefficients by using the twisted coefficients method. This method also works for constructing Khovanov homology for "non-oriented virtual knots" in the sense of [Viro], in particular, for knots in RP 3 .Virtual knots were introduced in mid-nineties by Lou Kauffman, see [KaV]. By a virtual diagram we mean a four-valent graph on the plane endowed with a special structure: each crossing is either said to be classical (in this case one indicates which pair of opposite edges at this crossing forms an overcrossing; the remaining two edges form an undercrossing) or virtual (in this case, we do not specify any addtional structure; virtual crossings are just marked by a circle). Two virtual diagrams are called equivalent if one of them can be obtained from the other by a finite sequence of generalized Reidemeister moves and planar isotopies. Recall that the list of generalized Reidemeister moves consists of classical Reidemeister moves (see,e.g., [Man]) and the detour move. The latter means that if wa have a purely virtual arc containing only virtual crossings, we may remove it and restore in any other place of the plane, see Fig. 1.

The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of 'realisable' graphs. When passing to generic graphs with the same equivalence relations we get 'graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalisation of the Kauffman-Murasugi-Thistlethwaite theorem on 'minmal diagrams' for graph-links.

We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the minimal classical crossing number for classical knots.Such projections can be useful for lifting invariants from classical knots to virtual knots. Different maps satisfy different properties.MSC: 57M25, 57M27