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Tropical Embeddings of Metric Graphs

Every graph $Γ$ can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number $\text{cross}\left(Γ\right)$. In this paper, we prove that if $Γ$ is a metric graph it can be realized as a tropical curve in the plane with exactly $\text{cross}\left(Γ\right)$ 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.