Tropical Embeddings of Metric Graphs

Adan Medrano Martin del Campo; Sylvain Carpentier

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arXiv · 1604.06176

Tropical Embeddings of Metric Graphs

Adan Medrano Martin del Campo·Sylvain Carpentier

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Abstract

Every graph $\Gamma$ can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number $\text{cross}\left(\Gamma\right)$. In this paper, we prove that if $\Gamma$ is a metric graph it can be realized as a tropical curve in the plane with exactly $\text{cross}\left(\Gamma\right)$ crossings, where the tropical curve is equipped with the lattice length metric. Our result has an application in algebraic geometry, as it enables us to construct a rational map of non-Archimedean curves into the projective plane, whose tropicalization is almost faithful when restricted to their skeleton.

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Adan Medrano Martin del Campo, Sylvain Carpentier. 2016-04-21. Tropical Embeddings of Metric Graphs. https://arxiv.org/abs/1604.06176

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