The space of immersed polygons

Abstract

We use the Schwarz-Christoffel formula to show that for every $n\ge 3$, 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 $n\ge 5$ by González and Sedano-Mendoza.

v1: 7 pages, 2 figures. v2: fixed some typos