A polynomial of graphs on surfaces
We consider ribbon graphs, i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a four-variable polynomial invariant of these objects, the ribbon graph polynomial, which has all the main properties of the Tutte polynomial. Although the ribbon graph polynomial extends the Tutte polynomial, its definition is very different, and it depends on the topological structure in an essential way.
Bollobás, B., & Riordan, O. (2002). A polynomial of graphs on surfaces. Mathematische Annalen, 323 (1), 81-96. https://doi.org/10.1007/s002080100297