A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($ 90^{\circ}$) and that of interior concave angles ($ 270^{\circ}$) is always $4$.
Consider a simple rectilinear polygon on a discrete torus ($m\times n$ grid with periodic boundary conditions). If it does not wind around the torus, then the above statement holds. If instead it winds around the torus, then the difference described above becomes $0$.
Does anyone have a clue how to prove this? I guess it has something to do with the winding number of the two polyliness which defines such polygon?