Partial duality of hypermaps

TL;DR

This paper introduces the concept of partial duality for hypermaps, generalizing classical duality, and provides combinatorial descriptions and genus change formulas for this new operation.

Contribution

It defines partial duality in multiple hypermap models and derives a genus change formula, expanding the understanding of hypermap dualities.

Findings

Partial duality generalizes classical Euler-Poincaré duality.

Provides combinatorial descriptions in three hypermap models.

Derives a formula for genus change under partial duality.

Abstract

We introduce partial duality of hypermaps, which include the classical Euler-Poincar\'e duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or $\tau$-model), or as three permutations on the set of half-edges (rotation system or $\sigma$-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.