Partial duality for ribbon graphs, I: Distributions

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 Bollobas–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.

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