Kronheimer on “Knot Groups and Lie Groups”

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 G, then the set of homomorphisms from the knot group of a knot K (the fundamental group of the complement of K) 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 G 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 G-bundle) sits in the space of all connections on a G-bundle.

To make everything concrete, let’s fix our group to be \mathrm{SO}(3), 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 \mathrm{SO}(3) on the 2-sphere?  And what does this tell us about how Khovanov homology should generalize?

12 thoughts on “Kronheimer on “Knot Groups and Lie Groups”

  1. Hi Ben, two questions:

    Why do you ask the second last question about “the connection of Khovanov homology to the action of SO(3) on the 2-sphere”? What does this have do with Kronheimer’s talk?

    Is this Kronheimer work at all related to Reshetikhin and Kashaev’s work on invariants of links with a flat connection in the compliment? It seems that they have an invariant come from one flat connection, whereas this Kronheimer invariant is about the space of all flat connections.

  2. A couple replies to Joel’s question. First off, the KaRe invariants (so as not to confuse them with Khovanov-Rozansky) use a complex lie group like SL(2), whereas Kronheimer is using a compact lie group like SO(3). Secondly, although KaRe invariants take a flat connection as input they only depend on the *gauge class* of the flat connection. That is the input really lies in Hom(pi_1, SL_2)/SL_2 (where the action is by conjugation). This is a smaller space that what Kronheimer is looking at (he only cares about the conjugacy class of the meridian), but the gauge classes are a refinement of the conjugacy classes.

  3. remember that we had to pick a conjugacy class, and I chose 180 degree rotations. These form an \mathbb{R}P^2. So, in fact, maybe I wanted \mathrm{SU}(2) and to take the lift of 180 degree rotations. This would give me a 2-sphere, and \mathrm{SU}(2) acts in the obvious way. So, one can think of the quandle in question as the natural quandle attached to any symmetric space: each point is goes to the reflection through it. Stated like this (at least to me), it sounds very suggestive that Khovanov homology is somehow tied to the geometry of this symmetric space.

  4. Noah (or anyone else),

    How are gauge classes related to conjugacy classes in general? I had thought gauge equivalence was given by changes of trivialization of a G-torsor, which (in the connected, simply connected case) are controlled by some kind of conjugation action by global sections. Am I confusing flat sections with zero sections or something?

  5. Scott: A gauge field is a connection on a G-bundle. It is often described relative to a trivialization (at least a local one). If you have such a trivialization, or even if you don’t, you can cosmetically twist a connection by multiplying the bundle in each by a map from the base M to the group G. In particular the map may be constant, which amounts to acting by conjugation. If the connection is flat, it is the conjugation action if you view the connection as a map from pi_1(M) to G.

    In this picture there is no such thing as a zero section. Not canonically anyway. If G is a connected, simply connected Lie group, then the connections are an affine space of the vector space of g-valued 1-forms. If you pick an origin, such as one afforded by a trivialization, then that marked origin is a cosmetic zero section.

  6. Scott: If you speak groupoid there’s a great way to view this connection.

    We’re really interested in the fundamental groupoid of the complement, and representations of that groupoid into some group G. Each such representation is just a way of assigning a group element to every homotopy class of paths so that composition is preserved. For definiteness, let’s consider SO(3,\mathbb{R}).

    Now we can think of some copy of \mathbb{R}^3 at each point of the space, and the representation tells us what transformation takes us from one copy of \mathbb{R}^3 to another along a given path. So what representations are “equivalent”? The ones that differ by changes of bases!

    So, to every point you’ve got a copy of GL(3,\mathbb{R}) you can use to change its basis. How does this affect a representation? Well if you’ve got a path \gamma:x\rightarrow y represented by R(\gamma)\in SO(3,\mathbb{R}), then after we change bases it’s represented by g_x^{-1}R(\gamma)g_y, where g_x and g_y are the change-of-basis matrices at x and y respectively. When we impose a suitable smoothness condition, these transformations are what we normally call “gauge transformations”.

    Okay, that looks complicated. But then you notice that the complement of the knot is connected, so its fundamental groupoid can be “skeletonized” — it’s equivalent to a group. We just pick one point and use the fundamental group with that base-point. Now we only have one object in our category, so only one copy of \mathbb{R}^3 to change basis with, and we’re back to regular old conjugations.

  7. just a comment about ben webster’s comment about symmetric spaces as quandles: when the symmetric space is a “bounded symmetric domain” you actually get a “commuting family” of quandle structures parameterized by the unit circle, i think.

  8. Thanks, John and Greg! I have more questions, if you don’t mind.

    First, when people mention 3-manifold invariants from gauge theory, I think “Chern-Simons” but those names haven’t shown up here. Is Kronheimer referring to another gauge theory on 3-manifolds whose stationary phases are flat connections?

    Second, when someone says “instanton” I think of classical solutions to the equations of motion, but I’ve also heard the word used to refer to solutions with localized singularities. Are the singularities just a way of encoding sections of a nontrivial bundle?

    Last, if I decide to work with a connection that is not flat (e.g. some Yang-Mills solutions IIRC), I should get an element of something for any homotopy between paths (and maybe for any n-simplex), but I’m not sure what that something is. If I view the curvature of the connection as a Lie(G)-valued 2-form, integrating over a disc seems to give me an element of Lie(G). If I parallel transport around the boundary, I get an element of G/G (conjugation action). Am I making a mistake?

  9. To your first question, I would say that he is looking at this moduli space of representations in part because it is the classical solutions to Chern-Simons, but in this talk he wasn’t doing any gauge theory. I am pretty sure that when he talked about gauge theory hiding the background, he did mean Chern-Simons theory (unless he had some 4-d thing in mind, but that’s definitely beyond the scope of the talk).

  10. Are you sure about instanton localized singularity? I was under the impression it meant compact support, that is localized anything. It’s an interaction that does nothing in the past, nothing in the future, nothing outside of a compact area of space, and is gone.. in an instant. To be honest, instantons get close enough to analysis that I haven’t really got more than the rough outlines.

    As for your last question, have you read Baez and Schreiber’s “Higher Gauge Theory”? Instead of representing the fundamental groupoid, they represent the fundamental 2-groupoid into a Lie 2-group, which allows exactly this sort of extension.

  11. Is Kronheimer referring to another gauge theory on 3-manifolds whose stationary phases are flat connections?

    I am not entirely sure, but I believe that instanton homology refers to the original Floer homology that Floer had in mind. Floer’s idea, in full generality, was to define Morse homology for Morse-like functions on infinite-dimensional manifolds. In particular, the Chern-Simons functional is such as function, on the manifold (indeed, affine vector space) of connections on a G-bundle on a 3-manifold M. So, in Morse homology in general, you construct the differential using the downward gradient flow. Again, I am not entirely sure, but I recall that a gradient flow of Chern-Simons gives you an anti-self-dual connection on M x R. And I think that this was the situation that Floer studied the most in developing his homology theory.

    So this is and isn’t Chern-Simons. It is a use of the Chern-Simons functional. The functional is not used as the phase of a path integral as it is in Chern-Simons field theory; rather it is used for an infinite-dimensional Morse homology. But the two uses are plausibly related.

  12. Greg,

    The bit about generating anti-self-dual connections on M x R sounds very interesting. I’m not sure if this describes an actual correspondence with what you said, but Yang-Mills instantons (ASD connections whose curvature at the boundary vanishes) naturally produce Chern-Simons solutions on the boundary, and the CS invariant of the boundary connection doesn’t depend on the interior configuration. Unfortunately, the boundary in this case is two oppositely oriented copies of M rather than M itself.

    I found this as a brief comment on the wikipedia article on instantons in a slightly mangled form, but it’s also in Donaldson-Kronheimer. Oddly enough, the wikipedia comment was there in the first revision, and the rest of the article seems to have been built around it.

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