ON BALANCED PLANAR GRAPHS, FOLLOWING W. THURSTON

Abstract : Let f : S 2 → S 2 be an orientation-preserving branched covering map of degree d ≥ 2, and let Σ be an oriented Jordan curve passing through the critical values of f . Then Γ := f −1 (Σ) is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such Γ, where f has 2d−2 distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.
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Sarah Koch, Tan Lei. ON BALANCED PLANAR GRAPHS, FOLLOWING W. THURSTON. 2015. 〈hal-01117927〉

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