Partial duality for ribbon graphs, II: Partial-twuality polynomials and monodromy computations

The partial (Poincaré) dual 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. In developing the theory of maps, Wilson and others have composed Poincaré duality ∗ with Petrie duality × to give Wilson duality ∗×∗ and two trialities ∗× and ×∗. In further expanding the theory, Abrams and Ellis-Monaghan have called the five operators twualities. Part I of this investigation (Gross et al., 2020) introduced as a partial-∗ polynomial of G, the generating function enumerating partial Poincaré duals by Euler-genus. In this sequel, we introduce the corresponding partial-×, -∗×, -×∗, and -∗×∗ polynomials for their respective twualities. For purposes of computation, we express each partial twuality in terms of the monodromy of permutations of the flags of a map. We analyze how single-edge partial twualities affect the three types (proper, untwisted, twisted) of edges. Various possible properties of partial-twuality polynomials are studied, including interpolation and log-concavity; machine-computed unimodal counterexamples to some log-concavity conjectures from Gross et al. (2020) are given. It is shown that the partial-∗×∗ polynomial for a ribbon graph G equals the partial-× polynomial for G^∗. Formulas or recursions are given for various families of graphs, including ladders and, for Wilson duality, a large subfamily of series–parallel networks. All of these polynomials are shown to be log-concave.