Ribbon graphs and bialgebra of Lagrangian subspaces

Abstract

Abstract. To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

Delta-Matroids and Vassiliev Invariants

“…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 .…”

“…In [11] it is shown that the first and the second Vassiliev moves for embedded graphs can be naturally expressed as base changes in the 2|E|-dimensional symplectic space over F 2 spanned by the edges of the graph and their duals. We reproduce the definition of these base changes below and show that it is compatible with the above definition of the Vassiliev moves for binary delta-matroids.…”

“…In [11] the action of Vassiliev moves on Lagrangian subspaces in V E was used to introduce the 4-term relations and the Hopf algebra of Lagrangian subspaces modulo the 4-term relations. There is a mapping taking an embedded graph to a Lagrangian subspace, and linear functionals on this Hopf algebra determine weight systems.…”

“…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].…”

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

“…In [9] Vassiliev moves were extended to framed diagrams, which are chord diagrams associated to ribbon graphs with possibly twisted ribbons, and the corresponding four-term relations were described. Kleptsyn and Smirnov in [7] extended Vassiliev moves to Lagrangian subspaces. Let, as above, E be a finite set, V E be the vector space over F 2 spanned by the elements of the set E E ∨ , and let e, e ∈ E be two distinct elements in E. Then the first Vassiliev move, assigned to a pair e, e , is a linear mapping V E → V E preserving all the basis vectors except for the vectors e ∨ , e ∨ .…”