On hyperspaces of knots and planar simple closed curves

Abstract

We consider the Vietoris hyperspaces $\mathcal{S}({\mathbb{R}}^n)$ of simple closed curves in ${\mathbb{R}}^n$, $n=2,3$, and their subspaces ${\mathcal{S}}_P({\mathbb{R}}^2)$ of planar simple closed polygons, ${\mathcal{K}}_P$ of polygonal knots, and ${\mathcal{K}}_T$ of tame knots. We prove that all the hyperspaces are strongly locally contractible, arcwise connected, infinite-dimensional Cantor manifolds, and $\mathcal{S}({\mathbb{R}}^2)$ and ${\mathcal{K}}_T$ are strongly infinite-dimensional Cantor manifolds. Moreover, ${\mathcal{S}}_P({\mathbb{R}}^2)$ and ${\mathcal{K}}_P$ are $\sigma$-compact, strongly countable-dimensional absolute neighborhood retracts.