So, I’m in lovely Edinburgh, Scotland (everyone I’ve told about this said “Scotland? In November?” but it’s not actually worse than New Jersey) in advance of the Maxwell Colloquium on Knot Homology.
By sheer luck, my trip here happened to overlap with the University of Edinburgh’s Whittaker Lecture which is a bit like the Bowen Lectures at Berkeley, except that there’s only one of them. By even more luck, the speaker with Prof. Peter Kronheimer (from Harvard) and the topic was “Knot Groups and Lie Groups.”
The talk was mostly elementary, though as most of you can guess if you look at some of Kronheimer’s papers, things like the self-dual Yang-Mills equation and Floer homology popped up from time to time (though in a mostly untroubling way).
He started with the simple observation that if you fix a group , then the set of homomorphisms from the knot group of a knot (the fundamental group of the complement of ) to your fixed group is a knot invariant. Better yet, one can break up the set of homomorphisms by what conjugacy class any meridian of the knot lands in (a meridian is a small loop around the knot. These are the generators of the fundamental group in the Wirtinger presentation, and thus are all conjugate). You could think of this as representations of the fundamental quandle into this conjugacy class, thought of as a quandle, but I’ll leave that to John Armstrong, and stick with groups.
If your group is finite, then all you have to do is count, and you will get something interesting. In fact, it was these cases that led people to consider quandles in the first place. But, as you all may have noticed, as much as I like finite groups, I think Lie groups are vastly superior, so maybe we should think about representations into Lie groups.
Now, there are a lot of different games you can play with topological spaces, to extract invariants from them. Unfortunately, the one Kronheimer was interested in is not one you’re likely to have heard of, or that I am qualified to explain. Let’s just say that it’s called instanton homology, and is a subquotient of the homology of this moduli space of representations, somehow determined by how this space of representations (thought of as flat connections in a -bundle) sits in the space of all connections on a -bundle.
To make everything concrete, let’s fix our group to be , and our conjugacy class to be the 180 degree rotation around a line.
So, you may all be wondering where I’m going with this (I certainly am), but at this point something a little surprising happens. You compute this instanton homology for a few small knots, and look over their other invariants on the Knot Atlas, and you notice “Huh, this looks just like Khovanov homology in a bunch of these cases.”
Now, if you’re the first person to do this, you think “Well, maybe it’s the same as Khovanov homology!” Then, you get very excited, until you reach the(3,5)-torus knot and discover that this is not, in fact the case. At this point, you would say something not appropriate for a family blog, and say “So, what is going on?”
This is unfortunately about the point the lecture reached, with the conclusion “Nobody really knows,” which is both a disappointing (because you want to know) and slightly exciting (because maybe you can figure it out) way to end a lecture.
Hopefully, it can serve the same function for a blog post. I don’t have anything else to say on the topic (at the moment), though I can’t help but wonder: What’s the connection of Khovanov homology to the action of on the 2-sphere? And what does this tell us about how Khovanov homology should generalize?