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A θ-curve is called almost trivial if it does not contain any non-trivial knot. A θ-curve is called strongly almost trivial if it has a planar projection which does not contain a projection of any non-trivial knot. In this paper, we introduce a method to present strongly almost trivial θ-curves. We also give an almost trivial θ-curve which may not be strongly almost trivial.

In the present paper, we construct Khovanov homology theory with arbitrary coefficients for arbitrary virtual knots. We give a definition of the complex, which is homotopy equivalent to the initial Khovanov complex in the classical case; our definition works in the virtual case as well. The method used in this work allows us to construct a Khovanov homology theory for "twisted virtual knots" in the sense of Bourgoin and Viro [4, 27] (in particular, for knots in RP³). We also generalize some results of the Khovanov homology for virtual knots with arbitrary atoms (Wehrli and Kofman–Champanerkar spanning tree, minimality problems, Frobenius extensions) and orientable ones (Rasmussen's invariant).

In this paper, we give a new proof of Vassilievs planarity criterion of a graph with vertices of degree 4 and an additional cross structure. It is also established that all height atoms are orientable. The criterion for an atom to be an orientable one is established in terms of its $f$-graph.