Exact Borel classification of tame and wild knot hyperspaces

Determine the exact Borel classes of the hyperspaces of tame knots (\(\mathcal K_T\)) and wild knots (\(\mathcal K_W\)) as subsets of the Vietoris hyperspace \(C(\mathbb{R}^3)\) of all continua in \(\mathbb{R}^3\), beyond the established facts that \(\mathcal K_T\) is Borel but not \(G_{\delta\sigma}\) and \(\mathcal K_W\) is Borel but not \(F_{\sigma\delta}\).

Background

The paper studies Vietoris hyperspaces of simple closed curves and knots, focusing on spaces such as $\mathcal S(\mathbb{R}^2)$, $\mathcal S_P(\mathbb{R}^2)$, and the knot hyperspaces $\mathcal K$, $\mathcal K_P$, $\mathcal K_T$, and $\mathcal K_W$. It establishes local contractibility, connectedness, and Cantor manifold properties for these spaces and determines the exact Borel class for $\mathcal S_P(\mathbb{R}^2)$ and $\mathcal K_P$ (both are $F_\sigma$ and not $G_\delta$).

In contrast, while $\mathcal K_T$ and $\mathcal K_W$ are shown to be Borel subsets of $C(\mathbb{R}^3)$, their precise positions within the Borel hierarchy are left unresolved; the paper notes only partial information ($\mathcal K_T$ is not $G_{\delta\sigma}$ and $\mathcal K_W$ is not $F_{\sigma\delta}$). Clarifying their exact Borel classes remains an explicitly stated open problem in the introduction.