A topological framework for signed permutations

In this paper we present a topological framework for studying signed permutations and their reversal distance. As a result we can give an alternative approach and interpretation of the Hannenhalli-Pevzner formula for the reversal distance of signed permutations. Our approach utlizes the Poincar\'e dual, upon which reversals act in a particular way and obsoletes the notion of "padding" of the signed permutations. To this end we construct a bijection between signed permutations and an equivalence class of particular fatgraphs, called $\pi$-maps, and analyze the action of reversals on the latter. We show that reversals act via either slicing, gluing or half-flipping of external vertices, which implies that any reversal changes the topological genus by at most one. Finally we revisit the Hannenhalli-Pevzner formula employing orientable and non-orientable, irreducible, $\pi$-maps.