The space of immersed polygons

We use the Schwarz-Christoffel formula to show that for every \(n\geq 3\), the space of labelled immersed \(n\)-gons in the plane up to similarity is homeomorphic to \(\mathbb{R}^{2n-4}\). We then prove that all immersed triangles, quadrilaterals, and pentagons are embedded, from which it follows th...

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Published inarXiv.org
Main Author Maxime Fortier Bourque
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 13.06.2024
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Summary:We use the Schwarz-Christoffel formula to show that for every \(n\geq 3\), the space of labelled immersed \(n\)-gons in the plane up to similarity is homeomorphic to \(\mathbb{R}^{2n-4}\). We then prove that all immersed triangles, quadrilaterals, and pentagons are embedded, from which it follows that the space of labelled simple \(n\)-gons up to similarity is homeomorphic to \(\mathbb{R}^{2n-4}\) if \(n\in \{3,4,5\}\). This was first shown by Gonzáles and López-López for \(n=4\) and conjectured to be true for every \(n\geq 5\) by González and Sedano-Mendoza.
ISSN:2331-8422