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They draw nice pictures of Riemann Surfaces in "Automatic Generation of Riemann Surface Meshes" by Matthias Nieser, Konstantin Poelke, and Konrad Polthier:

Riemann surface with 3 layers which are connected... and a more complicated example

Identify each layer with a vertex, then for the left it is easy to relate the Riemann surface with a graph: it is the cyclic graph with 3 vertices.

I'll try to explain: Think of the layers extended to infinity. Identify the a vertex with a point on a layer infinitely far way. Two points/vertex are connected if there is a transition between the two layers where the points live.

But how for more complicated Riemann surfaces like the right one? Is it (the other way round now) always possible to construct a Riemann surface for a given graph?

draks ...
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    It is unclear to me how you are associating a graph to a Riemann surface: Riemann surface do not come equipped with a system of cuts, contrary to the suggestive figures. There are some constructions in the literature of approximating Riemann surfaces by graphs so that, say, the spectra of Laplacian converge, but I do not think this is what you are after. – Moishe Kohan Dec 07 '16 at 07:42
  • @MoisheCohen, I'll try to explain: Think of the layers extended to infinity. Identify the a vertex with a point on a layer infinitely far way. Two points/vertex are connected if there is a transition between the two layers where the points live. But I'm not sure if this is too naive... – draks ... Dec 07 '16 at 07:51
  • You mean "ends" of the surface when you say layers. You still need impose some additional structure to get your transitions between ends. Your real pictures are embedded minimal surfaces which necessarily are stacked. – Charlie Frohman Dec 16 '16 at 04:40
  • A planar curve can be thought of as a Riemann surface, by projection on one variable. It has an amoeba, which might be what you are looking for. http://homepages.math.uic.edu/~jan/mcs595s14/curves.pdf – Charlie Frohman Dec 16 '16 at 05:10
  • @CharlieFrohman hmm, I remember reading about tropical geometry but never worked with amoeba.If you dare to work it out a bit as answer... – draks ... Dec 16 '16 at 07:30
  • @MoisheCohen did my explanation help? I ask because several others also complained... – draks ... Dec 16 '16 at 12:57
  • @draks... nope. – Moishe Kohan Dec 16 '16 at 16:14

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