Ribbon graphs and meromorphic functions

B. Shapiro

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arXiv · 2604.21358

Ribbon graphs and meromorphic functions

B. Shapiro

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Abstract

Let Y be a compact Riemann surface, phi:Y -> CP^1 a meromorphic function, and Gamma in Y a ribbon graph avoiding the critical points of phi. Then phi(Gamma) is an immersed graph in CP^1. Conversely, given an immersion im:Theta to bCP^1 of an abstract multigraph Theta without vertices of valence 1 or 2, we describe a construction of a compact Riemann surface Y and a meromorphic function phi_{im}:Y in CP^1 such that phi_{im}(Gamma)=im(Theta). We investigate the relation between the topology of Y and the combinatorics of Gamma. In particular, for a surface of genus g we construct spanning ribbon graphs whose underlying abstract graphs have arbitrary prescribed graph genus g' smaller or equal g, including the planar case. As a consequence, the number of self-intersections of ϕ(Gamma) cannot, in general, be controlled solely by the genus of Y. We establish general lower bounds for the number of self-intersections and formulate several open problems, with emphasis on planar ribbon graphs.

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BibTeX RIS

B. Shapiro. 2026-04-23. Ribbon graphs and meromorphic functions. https://arxiv.org/abs/2604.21358

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