Graphs on surfaces via planar graphs

I would like to present a joint work with Clark Butler [2] about a relations between some polynomial invariants of graphs on surfaces and planar graphs.
A famous graph invariant, the Tutte polynomial, was generalized to topological setting of graphs on surfaces by B. Bollobás and O. Riordan in [1] and to relative plane graphs by Y. Diao and G. Hetyei in [3]. We found a relation between these polynomials for graphs obtained by the construction below.
Graphs on surfaces can be studied in terms of plane graphs via their projections preserving the rotation systems. For non-planar graphs such a projection will have singularities. The simplest singularities are double points on edges of the graph. Using them we supplement the image of the graph with some additional edges and vertices. Thus we obtain a relative plane graph which is a plane graph with a distinguished subset of edges.
This relation has an application in knot theory. The classical Thistlethwaite theorem relates the Jones polynomial of a link to the Tutte polynomial of a plane graph obtained from a checkerboard coloring of the regions of the link diagram. Our relation conforms two generalizations of the Thistlethwaite theorem to virtual links from [3,4].