Systolic embedding of graphs on translation surfaces

2025/07/12 by Dey, Achintya, Sanki, Bidyut
#30F10 #32G15 and 53C22 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.09182

Abstract

An embedding of a graph on a translation surface is said to be 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 cellular (respectively 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≥2, 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.

Citations

• The maximum number of systoles for genus two Riemann surfaces with abelian differentials
• Graphs of Systoles on hyperbolic surfaces
• Translation surfaces and their orbit closures: An introduction for a\n broad audience

Related

• Translation surfaces with large systoles
• Graphs of Systoles on hyperbolic surfaces
• Existence of Circle Packings on Translation Surfaces
• Trace Systoles and Sink Constant
• Computing the second and third systoles of a combinatorial surface