On a Hopf Algebra in Graph Theory

Abstract

We introduce and start the study of a bialgebra of graphs, which we call the 4-bialgebra, and of the dual bialgebra of 4-invariants. The 4-bialgebra is similar to the ring of graphs introduced by W. T. Tutte in 1946, but its structure is more complicated. The roots of the definition are in low dimensional topology, namely, in the recent theory of Vassiliev knot invariants. In particular, 4-invariants of graphs determine Vassiliev invariants of knots. The relation between the two notions is discussed.

Analogs of the Orthogonal, Hamiltonian, Poisson, and Contact Lie Superalgebras in Characteristic 2

“…Symmetric bilinear forms over Z/2 recently appeared in Lando's description of Vasiliev invariants [14].…”

Ribbon graphs and bialgebra of Lagrangian subspaces

“…In this section we first recall this construction and then provide its generalizations: the graph bialgebra, due to Lando [17], and the bialgebra of L-spaces, which is the main construction of this paper.…”

“…This notion generalizes the notion of the bialgebra of graphs, introduced by S. Lando in [17]. More specifically, we define a bialgebra structure on the vector space freely spanned by all possible Lagrangian spaces.…”

“…The four-term relation descends to the space of intersection graphs. In [17] it was shown that the space of framed graphs modulo the four-term relation still forms a bialgebra (while the space of framed chord diagrams does not).…”

Mutant knots and intersection graphs

“…Weight systems have a graph counterpart, so-called 4-invariants of graphs [13]. The knowledge that a weight system depends only on the intersection graphs does not guarantee, however, that it arises from a 4-invariant.…”