The space of immersed polygons

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 R²ⁿ⁻⁴. It follows that the space of labelled simple n-gons up to similarity is homeomorphic to R²ⁿ⁻⁴ if n∈ \3,4,5\, which confirms one more case of a conjecture of Gonz\'alez and Sedano-Mendoza.