Just a reminder: a knot in is wild if there is no homeomorphism where is a smooth (or p. l.) knot. So, how does one create these things?
This is certainly not an exhaustive list. But the reason I am writing up such a list is I am interested in the following question: for what knots is the following true: there is NO homeomorphism where but there IS a homeomorphism .
Note: the Gordon-Lucke Theorem says that this cannot happen for non-wild knots. That is, tame knots are determined by their complements.
This can happen for some wild knots though. In fact, it is possible for an everywhere wild knot to have a complement which is homeomorophic to a knot with just one wild point.
So, here we go:
1. Infinite product and their limits:
The limit part comes from the following type of construction:
Start with a finite product knot, put a “follow” torus around it, then put in more of the same knot, and a second follow torus which is a satellite of the first knot and repeat this process infinitely often, adding more satellite tori each time.
Then weather or not you get a standard infinite product knot or a knot that is wild at every point depends on how one slides the tori around. In each case, the knot comes from the infinite intersection of the tori.
If the tori are all slid to a single section of the first tori and arranged in order, one gets the classical infinite product. If the knots are “spread out” as one moves to the smaller tori, then one gets the “everywhere wild knot”. See Shilepsky’s paper for details on the second kind of knot and Bothe’s paper for the first kind. I relate the two kinds of knots (thanks for a good referee’s suggestion!) in my paper.
2. Infinite product of tangles.
These are knots like those in the upper right hand corner. Note that these types of knots CAN contain a wild arc in the following sense: given the knot built as the union of two arcs joined at their endpoints, at least one of the arcs is wild. In the infinite product case, the knot is the union of two tame arcs.
I don’t know what the implications of having homeomorphic complements is here.
3. Intersection of a nested set of tori.
Again, I don’t know if these are determined by their complements or not; my guess would be “no”. In my paper, I come up with some invariants for these types of knots.
4. Death spiral knots. (my term for these)
The suggestion for construction is given above. Basically, one starts with the first stage of a type of knot discussed in part 3. But then in the first construction cylinder, one deletes the left most returning arc cylinder (keeps the spanning one). Then cut the torus at the left disk, narrow the cylinder to the left of the left disk, shrink it and glue it to the disk corresponding to the start of the spanning cylinder. Do this in a way that the smaller tori align.
Now the resulting knot is the set of points that lie in an infinite number of cylinders.
I REALLY don’t know what is happening with the complements here.