The space of immersed polygons

2024/06/04 by Maxime Fortier Bourque, Bourque, Maxime Fortier · 1 citation
Engineering · #Architecture and Computational Design #FOS: Mathematics #Geometric Topology (math.GT)

paper · pdf · doi:10.48550/arxiv.2406.02519

openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the Schwarz-Christoffel formula to show that for every n≥ 3, the space of labelled immersed n-gons in the plane up to similarity is homeomorphic to 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 ℝ^2n-4 if n∈ \3,4,5\. This was first shown by Gonzáles and López-López for n=4 and conjectured to be true for every n≥ 5 by González and Sedano-Mendoza.

Cited by

• Morse theory and moduli spaces of self-avoiding polygonal linkages

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• K-theory and immersions of spatial polygon spaces
• Moduli space of simple polygons
• Real projective space as a space of planar polygons
• Dissecting the 2-sphere by immersions
• The topology of spaces of polygons