The crossing number of polynomial curve systems
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
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arXiv subjects
Jasmin Jörg
Publications and source records attributed to Jasmin Jörg.
Sparse curve systems have intermediate growth type
A system of simple closed curves on a surface of genus $g$ is said to be sparse if their average pairwise intersection number does not exceed one. We show that the maximal size of a sparse curve systems grows roughly like a function of type $c^{\sqrt{g}}$, with $c$ between $2$ and $81938$.
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Systoles on punctured spheres
We determine the maximal number of systoles among all spheres with $n$ punctures endowed with a complete Riemannian metric of finite area.
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Crossing number of curves on surfaces
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.
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