Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

Abstract

Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying the four-term relations. Weight systems have graph analogues, so-called $4$-invariants of graphs, i.e. functions on graphs that satisfy the four-term relations for graphs. Each $4$-invariant determines a weight system. The notion of weight system is naturally generalized for the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded graph. Recently, two approaches have been suggested to extend the notion of $4$-invariants of graphs to the case of combinatorial structures corresponding to embedded graphs with an arbitrary number of vertices. The first approach is due to V.~Kleptsyn and E.~Smirnov, who considered functions on Lagrangian subspaces in a $2n$-dimensional space over $\mathbb{F}_2$ endowed with a standard symplectic form and introduced four-term relations for them. On the other hand, the second approach, the one due to Zhukov and Lando, suggests four-term relations for functions on binary delta-matroids. In this paper, we prove that the two approaches are equivalent.