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

Abstract The partial (Poincare) 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 Bollobas–Riordan polynomials. In developing the theory of maps, Wilson and others have composed Poincare 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 Poincare 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.