The space of immersed polygons

TL;DR

This paper characterizes the space of labelled immersed polygons in the plane using the Schwarz-Christoffel formula, showing it is homeomorphic to Euclidean space for all n ≥ 3, and confirms embeddedness for small polygons.

Contribution

It proves the homeomorphism of the space of labelled immersed polygons to Euclidean space for all n ≥ 3 and establishes embeddedness for triangles, quadrilaterals, and pentagons.

Findings

The space of labelled immersed n-gons is homeomorphic to ^{2n-4} for all n .

All immersed triangles, quadrilaterals, and pentagons are embedded.

The homeomorphism was previously known only for specific cases and is now extended to all n .

Abstract

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 Gonz\'ales and L\'opez-L\'opez for $n=4$ and conjectured to be true for every $n\geq 5$ by Gonz\'alez and Sedano-Mendoza.