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Crossing number of curves on surfaces
Published 8 Mar 2024 in math.GT | (2403.05148v1)
Abstract: We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing number of up to 12 curves on a surface of genus 2 and prove that minimising systems are unique up to homeomorphisms of the surface and isotopies of curves.
- A. Bondy and U.S.R. Murty. Graph Theory. Graduate Texts in Mathematics. Springer London, 2008.
- Max Dehn. Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flächen. Acta mathematica, 69:135–206, 1938.
- Thurston’s Work on Surfaces (MN-48). Mathematical Notes - Princeton University Press. Princeton University Press, 2012.
- A Primer on Mapping Class Groups. Princeton Mathematical Series. Princeton University Press, 2012.
- Joshua Evan Greene. On curves intersecting at most once. arXiv preprint arXiv:1807.05658, 2018.
- Crossing lemmas for curves on surfaces, 2024. In preparation.
- Maryam Mirzakhani. Growth of the number of simple closed geodesics on hyperbolic surfaces. Annals of mathematics, 168(1):97–125, 2008.
- Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space. Geometry & Topology, 12(4):1883–1919, 2008.
- Topological designs. Geometriae Dedicata, 168(1):221–233, February 2013.
- Hugo Parlier. Kissing numbers for surfaces. Journal of Topology, 6(3):777–791, April 2013.
- Paul Schmutz. Riemann surfaces with shortest geodesic of maximal length. Geometric & Functional Analysis GAFA, 3(6):564–631, 1993.
- Paul Schmutz. Systoles on Riemann surfaces. manuscripta mathematica, 85(1):429–447, 1994.
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