I’m giving a talk tomorrow afternoon at the CSUSB Mathematics department. I will explain some of the stuff I’ve been working on, then shift gears to show how the general machinery can be applied to another mathematical landscape: knots. Sticking purely with knots is a bit restrictive, so I almost immediately skip ahead to the open version of knots, i.e. tangles. Luckily John Armstrong posted an article on arXiv in 2005, where he showed (among other very nice things) that tangle groups really are tangle invariants.
While I probably will not mention it tomorrow, another landscape that is being opened up (in the same sense that tangles open up knots) is Markov processes. Blake Pollard is doing exciting work on this front, and the same general machinery I’ll be talking about should have some interesting things to say about Markov processes as well.