More info about this project

Q: “Why hasn’t this been submitted to a journal yet?”

This project something I wrote based on my undergrad thesis. I have been meaning to go back and get it “submission-ready” for a while now, but that fell by the wayside when I started getting my PhD research up and running (I’m in a totally different field now).

I’m also slightly paranoid that the results are known already (though I was more or less unable to find a reference when I did my literature search back in 2020). It seems like something that should be folklore. My undergraduate advisors didn’t know of existing work in this direction, but then again, neither of them had worked with wild knots before.

What’s in the paper?

The executive summary is that the paper addresses certain cases where you can vs. can’t perform countably-many Reidemeister moves.

As motivation, I present two different curves whose diagrams both look like they “should” be unknots. However, only one of them is (this can be shown using a sort of “invariant” for tame knots).

This is strange, because it really looks like you should be able to unknot them both using Reidemeister moves. The problem is subtle, but maybe obvious once you see it. Essentially, for the wild knot, when you perform all the Reidemeister moves you end up “dragging” points in the ambient space along in such a way that as you undo the crossings you pull more and more points of the ambient space down toward a wild point. In the limit, you end up losing bijectivity of your ambient isotopy because at least a countable sequence of points from the ambient space get mapped to the wild point.

In the end, I state a conjecture about how for a certain class of diagrams, adding a fourth Reidemeister move that was suggested by Kye Shi might recover “diagrams equivalent by moves \(\iff\) ambient isotopic” in the case where countably-many moves are performed.

It would be cool to see further work on this, since as far as I can tell the theory of wild knots has gone largely untouched for the past half-century-or-more. It would also be nice to know if there are some obscure references for this sort of thing that I didn’t find during my literature review.