Valence-partitioned genus polynomials and their application to generalized dipoles

Calculations of genus polynomials are given for three kinds of dipoles: with no loops; with a loop at one vertex; or with a loop at both vertices. We include a very concise, elementary derivation of the genus polynomial of a loopless dipole. To describe the general effect on the face-count and genus polynomials of the operation of adding a loop at a vertex, we introduce imbedding types that are partitions of integers, specifically, partitions of the valences of the vertices at which loops are to be added. Adding a loop at a root-vertex changes the possible number of imbedding types from the number of partitions of the valence prior to adding the loop to the number of partitions of the valence afterward.