Moscow-Petersburg seminar on low-dimensional mathematics September 16, 2005, 16:00 V.Manturov (Moscow) Vassiliev's conjecture on planarity of singular links and finite-type invariants Recently, V.A.Vassiliev formulated the following conjecture: a graph with cross structutre at vertices is not realizable on the plane if and only if it contains two cycles without common edges having precisely one intersection point. We prove this conjecture and address the question on embeddability of such graphs into 2-surfaces with some natural extra condition. This problem allows a combinatorial reformulation in the terms of matrices: for a given Z_{2}-symmetric matrix, find a splitting of basis vectors into two subsets minimizing the sum of ranks of the corresponding matrices. Such functions occur to satisfy the 4T-relation and generalized 4T-relation thus being connected to finite-type invariants. Also, 4-valent graps with extra structure are very closed to virtual knots and, in some sense, to Kauffman bracket and Khovanov homology. Several unsolved problems will be proposed. --- http://www.pdmi.ras.ru/~lowdimma