Rolfsen asked the question as to whether any knot is topologically isotopic to the unknot. Where a topological isotopy is a continuous path in $\operatorname{Emb}(S^1,\mathbb{R}^3)$.

From now on I will use isotopy to mean topological isotopy as stated above. Note this is not the same as smooth isotopy which is the standard equivalence relation used in knot theory.

For example the trefoil knot here is isotopic to the unknot. An animation of this is shown in Jim Belk's answer to Which two knots are isotopic but not ambient isotopic?.

It is currently known that every PL knot is isotopic to the unknot and in fact any wild knot that is locally flat at at least one point is also isotopic to the unknot (informally — by picking a tame neighbourhood and isotoping everything outside it to a point).

Let $K$ be a knot in $\mathbb{R}^3$ and consider locally flat neighbourhood $(U,K\cap U)\cong (\mathbb{R}^3,\mathbb{R})$ then $K$ pierces a disc at each point $x\in K\cap U$. Thus a contender for a counter example was conjectured to have to fail a disc at each point. The Bing sling is such an example. It is interesting that this condition is not sufficient. An example of a knot that fails to pierce a disk at each point that also bounds a disc (and so is isotopic to the unknot) was constructed by Gilliman in Sequentially 1-ULC tori. It is still open as to whether the Bing sling is isotopic to the unkot.

The Bing sling is the limit of a sequence of nested tori of which only the first is unkotted. My question (probably stupid) is why one can't define an isotopy of the Bing Sling and the unknot by contracting this torus to its centre. That is we pick a family of homeomorphisms of 3-space that fixes the boundary of the first unkotted torus but shrinks everything inside to its centre. The Bing sling lies inside this torus so we define the isotopy as the family of maps whose image is the composition of the standard embedding of the Bing sling into the first torus composed with the shrinking homeomorphism that fixes the boundary of the torus.

This defines a pseudo isotopy (a continuous map $F: S^1\times I\rightarrow \mathbb{R}^3$ where $F(-,t)$ is an embedding for $t\in [0,1)$ but only continuous for at $t=1$). I'm struggling to see what the problem to extending the isotopy is. The issue here seems like $F(-,1)$ will somehow fail to be injective but i am failing to describe it presicely. I would appreciate any help — I'm sure I must be missing something trivial.

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