Research

My research has been on translating knot detection problems of Khovanov homology or Knot Floer homology into tractable computational problems. Examples:

  • My doctoral dissertation is on the ability to detect the connect-sum of two same-handed trefoils (colloquially called the granny knot) by knot Floer homology, which is the first known knot detection result of its kind for a connect-sum of knots. This amounts to ruling out it being a satellite knot, or a hyperbolic knot. For the former, this was done using theory, but for the latter, this was where I used computational techniques. This case requires studying the fibering monodromy, which is a pseudo-Anosov map, which can be studied by train tracks. I wrote a program in Sagemath that can encoded train tracks on the disk; and the program can also perform “folding operations” on these train tracks. With this, I was able to get a relatively small list of train tracks that can possibly be associated with the monodromy. I then used pen and paper, working on these train tracks, to explicitly classify all such maps. These are then all eventually ruled out using things like the Alexander polynomial. You can find the Github repository for this program here.
  • I proved that Khovanov homology and knot Floer homology, together with some mild assumptions can detect the torus knot T(2,7). This is an application of the computational techniques used in this paper. I wrote a program that can handle the computational aspect of this technique for general T(2,n) knots for arbitrary n. You can find the Github repository here.

Another area of interest of mine that I am currently exploring is applying some machine learning tools to analyze geometric data, such as knot data or data of train tracks that can be generated by my program.

  • I am building/training a neural network to predict knot invariants, focusing specifically on slice genus at the moment. This was inspired by the works of Mark Hughes et al., here, and here. I am trying to improve on these frameworks by introducing newer machine learning techniques such as transformers. You can find the Github repository here.