# TDA and knot theory

I'm new to topological data analysis, and I learned some basics of it including persistent homology and mapper. In this paper, authors suggest a method to detect circle $$S^{1}$$, which is 1-dimensional object but can't be embedded into $$\mathbb{R}$$. To resolve such problem, they give an algorithm to compute persistent cohomology $$H^{1}$$ and find a coordinate for circles. There are some examples containing a circle, trefoil knot, pair of circles, and torus.

Especially, I'm interested in the trefoil knot case: when we compute the persistence diagram or scatter plot, it is almost same as $$S^{1}$$ case, since they are homeomorphic (but not isotopic). However, some big difference is that the scatter plot of the trefoil knot has some bumps (bulges) corresponding to the three high-density regions of the sampled curve, which occur when the curve approaches the central axis of the torus. I understand this, but I think this won't be helpful if the knot is very complicated or even unknotted (but still complicated). So I want to know if there's a method to detect such knotted feature by a computational method.

Since complements of $$S^{1}$$ and the trefoil knot have different fundamental groups, maybe we can try to compute persistent fundamental group of given data clouds, and this paper seems to give an algorithm to compute fundamental groups. However, it doesn't give many examples, especially about classifying knots. Since there are a lot of knot invariants, there will be a better way to do computationally.

The first thing I want to do is discriminate unknot ($$S^{1}$$) and the trefoil knot. So for each space, we randomly choose a lot of points (like 1000) and give some noise on it (give coordinatewise noise by normal distribution or whatever) so that the result will be two point clouds, which looks similar to $$S^{1}$$ and trefoil knot respectively. If we compute persistent homology of these clouds, that will give the almost same result - there will be one persistent barcode corresponds to the nontrivial class in $$H_{1}$$. (I think cohomology would work, too.) So it is not useful to detect unknot.

So we can think about possible knot invariants that can discriminate $$S^{1}$$ and the trefoil knot. The first thing came up in my mind is $$\pi_{1}(\mathbb{R}^{3} \backslash K)$$ (or more precisely, $$\pi_{1}(S^{3} \backslash K)$$), the fundamental group of knot complement. As I said, we have a tool to compute persistent fundamental group, but it is not useful because we have data about our knots, not a knot complement. It seems possible to generate points in knot complement by using given data, but it would require a lot of time.

I think this may help in biology since it is known that DNA has some interesting non-trivial knot structure. (See here) In fact, as we can see in the article, they already study it by looking at crossing numbers. But since this is not a good invariant (as I know), using more delicate invariant will help us to discriminate more complicated knots.

• I've had good success finding circles with the Hough Transform; I'd commend that to your attention. It's well-situated for parallelization, as well. – Adrian Keister Jun 19 at 13:20
• Can you highlight your question? It's not clear to me exactly what you are asking, but it seems interesting – Andres Mejia Jun 19 at 21:22
• @AdrianKeister Could you elaborate more? Maybe you can give it as an answer. I don't know what is the Hough transform and parallelization. – Seewoo Lee Jun 20 at 1:49
• @AndresMejia I added more explanation. Hope this helps. – Seewoo Lee Jun 20 at 1:50
• @SeewooLee: Well, it's really not enough material for an answer. I've explained the Hough Circle Transform, and even given native LabVIEW code for it here: winemantech.com/blog/…. The basic idea is rather straight-forward, but the Hough Transform is robust with respect to how much of the circle is actually present in the data. – Adrian Keister Jun 20 at 13:19