I’m new and just playing around with graph embeddings on higher genus (at least 3, but not more than 7) orientable surfaces. I’d like to draw on flat paper.
I’m using rectangles for holes and colour-coded edges. It’s just very messy.
Is there a better, perhaps standard, way?
One way to draw an $n$-holed torus in the plane (or to construct one topologically) is to take a $4n$-sided polygon and identify opposite sides. When drawing a graph on that surface, your vertices are points inside your polygon, and your edges are free to go out of the polygon through one side and come back from the opposite side.
For example, here is an embedding of the Petersen graph in the $1$-holed torus:
You might be able to avoid too many weird edges by placing vertices on the sides of the polygon.