ИСТИНА

The talk is focused on some particular family of statistical models on 6-valent graphs and their relations with the theory of invariants of 2-knots, i.e. the isotopy classes of S2 embeddings in R4, and integrable 3-d models of statistical physics on regular latices. In both cases the integrability properties are provided by the higher homotopic algebraic structure related with the Zamolodchikov tetrahedron equation. I will briefly recall some facts related with the problem of constructing invariants of 2-knots. This issue is principally due to the work of Carter, J.S., Jelsovsky, D., Langford, L., Kamada, S., Saito, M., and is based on quandle cohomology. In our works (with G.I. Sharygin and I.G. Korepanov) we extend in some sense their approach using the algebraic structure underlying the tetrahedral equation. The role of the tetrahedral cohomology will be especially emphasized. Another interesting aspect of this construction is the integrability of the considered statistical model when restricted to the regular periodic 3dimensional lattice. I will comment on the relation of the subject with the famous Maillet result in 2-d lattices.