Extending graph invariants to embedded graphs

A key problem in the study of graph invariants is how to extend such an invariant to embedded graphs, that is, to graphs on surfaces. This question is studied in many recent papers. In most of these papers the invariant in question of embedded graphs is defined as the original invariant of the underlying abstract graph endowed with an appropriate extension encoding some information about the embedding. I, however, is mostly interested in extending weight systems corresponding to finite type knot invariants to weight systems corresponding to link (multicomponent knots) invariants. To a singular knot, a chord diagram is associated. In turn, chord diagrams can be interpreted as orientable embedded graphs with a single vertex. From this point of view, an embedded graph is associated to a given singular link, and the number of vertices in this graph equals the number of components of the link.
In a number of recent papers, an approach to extending weight systems and graph invariants to arbitrary embedded graphs, which is based on the study of the structure of corresponding Hopf algebras. The space of graphs, as well as the space of chord diagrams modulo 4-term relations are endowed with natural connected graded Hopf algebra structures. We do not know about such a structure on the space of embedded graphs; however, it exists on the space of binary delta-matroids, which are combinatorial objects encoding important information about the structure of a graph, or a chord diagram, as well as about an embedded graph. Several examples of extensions of graph invariants to embedded graphs and delta-matroids by means of the corresponding Hopf algebra structures will be given.