This isn’t actually what I’m focusing on, but my friend Matt does his research with this thing. I think it’s pretty cool so I’ll try to explain what it is. P.S. I signed up for a class called “Explaining Science” this semester so hopefully my exposition will get clearer. Or I just need to write more math posts and they will get better.
I’m going to chat with you about the curve complex, so first, let’s think about curves.
A simple closed curve on a surface is a big ol’ loop (closed) that doesn’t cross itself (simple). You can think of these as made of stretchy pieces of string. Two curves are homotopic if you can push the string from one curve to form the other curve, without jumping across holes.
So for our example, no matter how you move it, the blue string won’t ever look like either of the red ones, because it’s “stuck” on the big hole in the middle, and no matter how you manipulate it, it has to get around that hole by going to the left half of the torus. Versus the two red curves are homotopic, because you could suck in that big alien arm from the top red curve, and the little thumb wrapping around the bottom, and then you’d look like the bottom red loop. We say the red curves are both representatives of the same homotopy class.
At first, it might be tempting to think there’s only two homotopy classes of loops on the torus: red ones, and blue ones. But then you get green ones:
Our green guy wraps around the horizontal axis of our torus once, but it wraps around the vertical way four times. It totally can’t be manipulated to look like either the red or the blue curves. [Aside: knot theorists could call this a (4,1) torus knot, for obvious reasons, but since one of the numbers is a 1, the knot is trivial. E.g. the green curve is just a loop if you took away the torus. But if it was, say, a (4,2) knot, then it wouldn’t be trivial if you took away the torus.]
My point was this: there’s a whole bunch of homotopy classes of simple closed curves on this very simple surface. And on any surface, there’s gonna be a whole bunch of homotopy classes of simple closed curves. (In fact, infinitely many.)
Instead of looking at the surface, we can look at those curves and talk about them. But it’s pretty impossible to just look at a list of curves and say things about it, just like it’s way hard to look at a list of roads and figure out the best way to get from A to B. What we need is a road map to show how these homotopy classes interact with each other. And that road map is our curve complex.
So let’s make this map. For each homotopy class, we form a vertex (like a city). So in our example, we’re holding three vertices, say R, G, and B. Then we draw an edge between vertices (like a road between cities) if the homotopy classes have some representatives that don’t intersect.
In our case, no matter how you draw them, all three of our curves intersect. So they’re just three disjoint vertices in our curve complex.
Here’s a quick example of curves that do have disjoint representatives (image found here):
Here, the a curves are disjoint, so in the curve complex, there’s an edge in between them. Anytime we have n classes that have representatives that are pairwise disjoint, we add an n-simplex. So if we had three curves A,B,C so that A and B had pairwise disjoint representatives, B and C had pairwise disjoint representatives, etc., we’d have a triangle in our complex. Note that these are representatives, so even if the B curve that is disjoint from A intersects C, we’re OK if there’s another curve homotopic to that B curve which is disjoint from C.
So this is the object that my friend Matt thinks about. There’s a couple cool things to say about it, and I’ll do another post later about this paper I read, but here’s a few goodies:
The curve complex is locally infinite. So if you’re standing in one city, and you look around, there are infinitely many roads heading out to other cities. Another way of saying this is that there are infinitely many vertices that are distance 1 away from your vertex.
It’s “hard” to find curves far away from each other. If I give you a curve on a surface, you can probably draw a disjoint nontrivial curve; there’s so many distance 1 away. In order for two curves to be distance two apart, they have to intersect, and there has to be a third curve disjoint from both of them. In our picture above, intersect, and both are disjoint from , so that’s okay too.
So for two curves to be bigger than distance 2 apart, there can’t be a third curve disjoint from both of them. That means that if you had scissors and cut along one curve, and then cut all along the other, you’d end up with a pile of discs (so any curve disjoint from the first two would be trivial). Another way is saying that these guys have to fill the surface.
We just did distance 1, 2, 3. It gets harder from there.
But the curve complex has infinite diameter. I don’t know this well enough to explain it here. But it’s true: there are roads that have infinitely many cities hanging out on them in our map.
This seems like enough for now. Curve complexes are pretty hip, and the paper I read was just submitted in October by Joan Birman and William Menasco. This whole post is in the first paragraph of that paper, so I referred to this talk by Moon Duchin while writing. There’s a standard way to cite references in math blog posts, just like people standardly write recipes at the end of baking blog posts. I’ll get the hang of it. I put in the reference for the second picture by it, but the first picture is an MS Painted version of this picture by Paul Bourke.
Okay! Math blog post 2, done!