Invariants of Framed Graphs and the Kadomtsev—Petviashvili Hierarchy

Abstract

S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf algebra of graphs to the Hopf algebra of polynomials in infinitely many variables). They proved that, after an appropriate rescaling of the variables, the result of the averaging of almost every such invariant over all graphs turns into a linear combination of single-part Schur functions and, thereby, becomes a τ-function of an integrable Kadomtsev-Petviashvili hierarchy. We prove a similar assertion for the Hopf algebra of framed graphs. At the same time, we show that there is no such an analogue for a number of other Hopf algebras of a similar nature, in particular, for the Hopf algebras of weighted graphs, chord diagrams, and binary delta-matroids. Thus, it turns out that the Hopf algebras of graphs and framed graphs are distinguished among the graded Hopf algebras of combinatorial nature.