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In the study of Riemann surfaces, we can get lots of cool examples by identifying parallel sides of polygons. For example, by identifying the parallel sides of a square we obtain a torus. This is easy enough to imagine. However, take this figure, for example,

and identify the sides with the same color. (Ignore the dashed lines.) This is also a Riemann surface. But one that I cannot imagine. I would love to get some geometric intuition by understanding this surface.

Particularly, I am doing a project in which this surface appears very frequently. If I could draw a pretty image of this surface, it would make an awesome cover of the report. I would appreciate any hint about how I could do it.

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Actually, you need more information to uniquely define how to identify the sides of the polygon. That is, you need to specify in which of two directions the edges are to be identified, along or against each other. For example, the parallel sides of a square can be joined to form a cylinder, or against each other to form a Mobius strip.

It can be shown that any polygon with identified oriented edges is homeomorphic to one of (i) a torus with $n$ handles, or (ii) a projective plane with $n$ handles. The proof involves a procedure for simplifying such diagrams to "standard" ones. This is how it works for your diagram if we assume the edges are oriented along each other:

enter image description here

Cut along the diagonal, move the lower part so the red parts are identified, to get the last diagram. The top and bottom parts are clearly tori, which if you join along the colored edges gives a torus with two holes.

enter image description here

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