Systolic embedding of graphs on translation surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916840478867456 |
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| author | Dey, Achintya Sanki, Bidyut |
| author_facet | Dey, Achintya Sanki, Bidyut |
| contents | 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 $Σ_n$ of $n$ circles, $n\geq2$, we investigate that $Σ_n$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_09182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Systolic embedding of graphs on translation surfaces Dey, Achintya Sanki, Bidyut Geometric Topology 30F10, 32G15 and 53C22 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 $Σ_n$ of $n$ circles, $n\geq2$, we investigate that $Σ_n$ 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. |
| title | Systolic embedding of graphs on translation surfaces |
| topic | Geometric Topology 30F10, 32G15 and 53C22 |
| url | https://arxiv.org/abs/2507.09182 |