Partial duality for ribbon graphs, I: Distributions. (May 2020) - Gross, Jonathan L.; Mansour, Toufik; Tucker, Thomas W.
Abstract: The partial dual G A with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials, and it has developed into a topic of independent interest. This paper studies, for a given G, the enumeration of the partial duals of G by Euler genus, as represented by its generating function, which we call the partial-dual Euler-genus polynomial of G . A recursion is given for subdivision of an edge and is used to derive closed formulas for the partial-dual genus polynomials of four families of ribbon graphs. The log-concavity of these polynomials is studied in some detail. We include a concise, self-contained proof that χ ( G A ) = χ ( A ) + χ ( E ( G ) − A ) − 2 | V ( G ) |, where χ ( G ) = | V ( G ) | − | E ( G ) | + | F ( G ) |, and where A represents the ribbon graph obtained from G by deleting all edges not in A . This formula is a variant of a result of Moffatt.