The space of immersed polygons

Maxime Fortier Bourque

Published 2024-06-04Version 1

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 $\mathbb{R}^{2n-4}$. It follows that the space of labelled simple $n$-gons up to similarity is homeomorphic to $\mathbb{R}^{2n-4}$ if $n\in \{3,4,5\}$, which confirms one more case of a conjecture of Gonz\'alez and Sedano-Mendoza.