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Systolic embedding of graphs on translation surfaces
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Systolic embedding of graphs on translation surfaces
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Systolic embedding of graphs on translation surfaces
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Paper
Systolic embedding of graphs on translation surfaces
2025
Overview
An embedding of a graph on a translation surface is said to be \\emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be \\emph{cellular} (respectively \\emph{essential}) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge \\(\\Sigma_n\\) of \\(n\\) circles, \\(n\\geq2\\), we investigate that \\(\\Sigma_n\\) admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we have identified another rich collection of graphs with more than one vertex that also admit cellular-sytolic embedding on translation surfaces.
