The number of Pfaffian orientations on punctured polygonally cellulated surfaces

In this paper, we introduce the notion of Pfaffian orientations on (punctured) polygonally cellulated orientable surfaces, and provide an expression for the number of such orientations. This generalizes the notion of Pfaffian orientations on planar graph, where a planar graph is seen as a punctured $2$-sphere, embedded in ${\mathbb{R}}^3$. So, as a direct corollary of our main theorem, we derive the number of Pfaffian orientations on a planar graph.