Delta-Matroids and Vassiliev Invariants

Abstract

binary delta-matroids modulo the 4-term relations so that the mapping taking a chord diagram to its delta-matroid extends to a morphism of Hopf algebras. One can hope that studying this Hopf algebra will allow one to clarify the structure of the Hopf algebra of weight systems, in particular, to find reasonable new estimates for the dimensions of the spaces of weight systems of given degree. Also it would be interesting to find a relationship between the Hopf algebras arising in this paper with a very close to them in spirit bialgebra of Lagrangian subspaces in [11].The authors are grateful to participants of the seminar "Combinatorics of Vassiliev invariants" at the Department of mathematics, Higher School of Economics and Sergei Chmutov for useful discussions.A set system (E; Φ) is a finite set E together with a subset Φ of the set 2 E of subsets in E. The set E is called the ground set of the set system, and elements of Φ are its feasible sets. Two set systems (E 1 ; Φ 1 ), (E 2 ; Φ 2 ) are said to be isomorphic if there is a one-to-one map E 1 → E 2 identifying the subset Φ 1 ⊂ 2 E 1 with the subset Φ 2 ⊂ 2 E 2 . Below, we make no difference between isomorphic set systems.A set system (E; Φ) is proper if Φ is nonempty. Below, we consider only proper set systems, without indicating this explicitly.

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

“…To define the second Vassiliev move, they use the recently introduced (see [13]) concept of handle sliding for deltamatroids. In [10], it is shown (see Proposition 4.10) that the action of the first and second Vassiliev moves on the space V E as defined by Kleptsyn-Smirnov coincides with the one defined by Zhukov and Lando for binary delta-matroids. Taking into account Theorem 2.1, we obtain the following statement.…”

“…This multiplication can be naturally transferred to the vector space CL , spanned by the Lagrangian subspaces, considered up to renumbering finite element sets. Meanwhile, in [10], a graded Hopf algebra of binary delta-matroids is constructed…”

“…In [10], the authors define the first and the second Vassiliev moves for binary delta-matroids B E . To define the second Vassiliev move, they use the recently introduced (see [13]) concept of handle sliding for deltamatroids.…”

“…Meanwhile, Lando and Zhukov in [10] constructed a Hopf algebra of binary delta-matroids, introduced four-term relations for them and constructed a quotient Hopf algebra modulo the four-term relations. The correspondence between delta-matroids and embedded graphs allows one to associate a weight system to a linear functional on the latter Hopf algebra.…”

Skew characteristic polynomial of graphs and embedded graphs

“…Fortunately, however, a construction due to A. Bouchet allows one to associate to each graph and each embedded graph its delta-matroid. In turn, delta-matroids span a Hopf algebra [25], and one can define the skew characteristic polynomial for delta-matroids following the same pattern as above. The construction makes use of the fact that we know how to extend nondegeneracy to delta-matroids.…”