The space of immersed polygons
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...
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| Format | Journal Article |
| Language | English |
| Published |
04.06.2024
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| Online Access | Get full text |
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Bourque, Maxime Fortier |
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| BackLink | https://doi.org/10.48550/arXiv.2406.02519$$DView paper in arXiv |
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| Snippet | 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... |
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| SubjectTerms | Mathematics - Geometric Topology |
| Title | The space of immersed polygons |
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