Lagrangian subspaces, delta-matroids and four-term relations

TL;DR

This paper demonstrates the equivalence of two recent approaches to extending four-term invariants from graphs to embedded graphs and related combinatorial structures, linking Lagrangian subspaces and delta-matroids.

Contribution

It proves the equivalence between the approach based on Lagrangian subspaces and the delta-matroid approach for four-term invariants of embedded graphs.

Findings

Established the equivalence of the two approaches.

Unified the frameworks for four-term invariants.

Enhanced understanding of invariants in knot theory and graph theory.

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.