A face polynomial invariant for embedded graphs

Abstract

We introduce the face polynomial of a ribbon graph, a generating function for face numbers under the ribbon group action. It addresses the face-count case of the Ellis--Monaghan--Moffatt question on twisted duality. We derive deletion--contraction recursions and prove that this polynomial coincides with the vertex polynomial for \(\langleδ\rangle\), \(\langleδτ\rangle\), and \(\langleδ,τ\rangle\). We characterize monomial cases and obtain formulas and recursions for Euler-genus partial-twuality polynomials. For orientable graphs, we show that the partial-dual face polynomial has both parities whenever an edge exists, so the partial-dual Euler-genus polynomial has only even exponents.