I have a not-secret love affair with blogging the curve complex: I (intro), II (dead ends), III (connected). I’m surprised I didn’t blog the surprising and cute and wonderful proof that the curve complex is hyperbolic, which came out two years ago. Maybe I’ll do that next math post (but I have a large backlog of math I want to blog). Anyways, I was idly scrolling through arXiv (where mathematicians put their papers before they’re published) and saw a new paper by the two who did the dead ends paper, plus a new co-author. So I thought I’d tell you about it!
If you don’t remember or know what the curve complex is, you’d better check out that blog post I (intro) above (it is also here in case you didn’t want to reread the last paragraph). Remember that we look at curves (loops) up to homotopy, or wriggling. In this post we’ll also talk about arcs, which have two different endpoints (so they’re lines instead of loops), still defined up to homotopy.
The main thing we’ll be looking at in this post are geodesics, which are the shortest path between two points in a space. There might be more than one geodesic between two spaces, like in the taxicab metric. In fact, in the curve complex there are infinitely many geodesics between any two points.
Infinity is sort of a lot, so we’ll be considering specific types of geodesics instead. First we need a little bit more vocabulary. Let’s say I give you an arc and a simple (doesn’t self intersect) closed curve (loop) in a surface, and you wriggle them around up to homotopy. If you give me a drawing of the two of them, I’ll tell you that they’re in minimal position if the drawings you give me intersect the least number of times of all such drawings.
If you have three curves a, b, c all in minimal position with each other, then a reference arc for a,b,c is an arc which is in minimal position with b, and whose interior is disjoint from both a and c.Now if you give me a series of curves on a surface, I can hop over to the curve complex of that surface and see that series as a path. If the path $latex v_0,v_1,\ldots,v_n$ is geodesic, then we say it is initially efficient if any choice of reference arc for intersects at most n-1 times.
The geodesic is an efficient geodesic if all n of these geodesics are initially efficient: . In this paper, Birman, Margalit, and Menasco prove that efficient geodesics always exist if have distance at least three.
Note that there are a bunch of choices for reference arcs, even in the picture above, and at first glance that “bunch” looks like “infinitely many,” which sort of puts us back where we started (infinity is a lot). Turns out that there’s only finitely many reference arcs we have to consider as long as . Remember, if you’ve got two curves that are distance three from each other, they have to fill the surface: that means if you cut along both of them, you’ll end up with a big pile of topological disks. In this case, they take this pile and make them actual polygons with straight sides labeled by the cut curves. A bit more topology shows that you only end up with finitely many reference arcs that matter (essentially, there’s only finitely many interesting polygons, and then there are only so many ways to draw lines across a polygon).
So the main theorem of the paper is that efficient geodesics exist. The reason why we’d care about them is the second part of the theorem: that there are at most many curves that can appear as the first vertex in such a geodesic, which means that there are finitely many efficient geodesics between any two vertices where they exist.
Look at this picture! The red curve and blue curve are both vertices in the curve complex, and they have distance 4 in the curve complex, and here they are on a surface! So pretty!
If you feel like wikipedia-ing, check out one of the authors on this paper. Birman got her Ph.D. when she was 41 and is still active today (she’s 88 and a badass and I want to be as cool as she is when I grow up).