ИСТИНА

Cobordism in 1-dimensional case is a well-known and long-studied equivalence relation on classical knots. In a natural way it can be defined for virtual knots, and by the standard procedure involving Gauss diagrams --- for free knots. In turn, free knots may be interpreted as framed 4-valent graphs. Thus, a notion of {\em framed 4-graphs cobordism} arises. we consider graphs with one unicursal component. To be precise, we give the following definition: \begin{definition} Two framed graphs $\Gamma_1, \Gamma_2$ are said to be {\em cobordant} if there exists a triple $(M^3, S, f)$ where $M^3$ is a 3-manifold, $S_g$ is an orientable 2-surface and $f\colon S_g \to M^3$ such that \begin{itemize} \item the image of $f$ has only standard types of singularities (double lines, triple points, cusp points); \item f(\partial (S_g\setminus (D^2 \sqcup D^2)) = \Gamma_1 \sqcup \Gamma_2. \end{itemize} Minimal $g$ among the surfaces $S_g$ in such triples is called the {\em cobordism genus}. \end{defniition} \begin{definition} A framed 4graph $\Gamma$ is called {\em slice} if it is cobordant to the trivial graph given by a circle with no vertices. \end{definition} The question of sliceness for graphs is an interesting and important one. Recently the following sliceness criterion was proved (together with V.O. Manturov): \begin{theorem} Let $\Gamma$ be framed 4-graph such that every vertex of $\Gamma$ is odd in the sense of Gaussian parity. Then $\Gamma$ is genus zero slice if and only if there exists a pairing of the chords of its chord diagram without intersections. \end{theorem} This theorem is important for in case of odd framed 4-graphs it gives a finite combinatorial algorithm solving the problem of sliceness.