Oh, you’re bringing me into knot theory here with categorification! While I never was one for the math, I enjoy the theories and explanations of such.
First exposed to knot theory in 1990 (I was supposed to go into Quantum physics and I REALLY WANTED to learn about knot theory but ended up in child psych instead because the math professor was on sabbatical) – a quick search in Wikipedia shows me what I was looking for:
https://en.wikipedia.org/wiki/Khovanov_homology which I first heard about a few years ago because it could detect the Unknot.
As someone who seems to be forever untangling cords and Christmas lights and my brain and hands trying to figure out the most efficient ways to do so *as I’m doing it*, all 10 fingers engaged, my brain’s own “style of math” (which looks nothing like math scribble and more like lines and pressures and gradients of light), that the math ppl figured out HOW to get to the Unknot was pretty exciting to me.