Ribbon graphs and meromorphic functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911618280980480 |
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| author | Shapiro, B. |
| author_facet | Shapiro, B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21358 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ribbon graphs and meromorphic functions Shapiro, B. Algebraic Geometry Algebraic Topology 05C10, 14H15, 30F10, 57K10 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. |
| title | Ribbon graphs and meromorphic functions |
| topic | Algebraic Geometry Algebraic Topology 05C10, 14H15, 30F10, 57K10 |
| url | https://arxiv.org/abs/2604.21358 |