Rolfsen asked the question as to whether any knot is isotopic to the unknot.
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.
A contender for a counter example was conjectured to have to fail a disc at each point thus a contender would be the Bing Sling. However this condition is not sufficient as 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. 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.
I'm struggling to see what the problem here is. I feel there must be something wrong with the continuity at the end point of the isotopy but I am failing to describe it presicley. I would appreciate any help - I'm sure I must be missing something trivial.