I’ve mentioned Outside In before. That video shows a way of turning a sphere inside out. It’s simpler than the first explicit eversions to be discovered, but the simplicity is connected to a high degree of symmetry. This leads to very congested parts of the movie, where it’s very difficult to see what’s going on. Further, many quadruple points — where four sections of the surface pass through the same point — occur simultaneously, and even higher degree points occur. We need a simpler version.
What would constitute “simple” for us, then? We want as few multiple points as possible, and as few at a time as possible. In fact, it would be really nice if we could write it down algebraically, in some sense? But what sense?
Go back to the diagrammatics of braided monoidal categories with duals. There we could draw knots and links to represent morphisms from the monoidal identity object to itself. And topologically deformed versions of the same knot encoded the same morphism. This is the basic idea of the category of tangles.
But if we shift our perspective a bit, we consider the 2-category of tangles. Instead of saying that deformations are “the same” tangle, we consider explicit 2-isomorphisms between tangles. We’ve got basic 2-isomorphisms for each of the Reidemeister moves, and a couple to create or cancel caps and cups in pairs (duality) and to pull crossings past caps or cups (naturality). Just like we can write out any link diagram in terms of a small finite collection of basic tangles, we can write out any link diagram isotopy in terms of a small finite collection of basic moves.
What does a link diagram isotopy describe? Links (in our picture) are described by collections of points moving around in the plane. As we stack up pictures of these planes the points trace out a link. So now we’ve got links moving around in space. As we stack up pictures of these spaces, the links trace out linked surfaces in four-dimensional space. And we can describe any such surface in terms of a small collection of basic 2-morphisms in the braided monoidal 2-category of 2-tangles. These are analogous to the basic cups, caps, and crossings for tangles.
Of course the natural next step is to consider how to deform 2-tangles into each other. And we again have a small collection of basic 3-morphisms that can be used to describe any morphisms of 2-tangles. These are analogous to the Reidemeister moves. Any deformation of a surface (which is written in terms of the basic 2-morphisms) can be written out in terms of these basic 3-morphisms.
We can simplify our picture a bit. Instead of knotting surfaces in four-dimensional space, let’s just let them intersect each other in three-dimensional space. To do this, we need to use a symmetric monoidal 3-category with duals, since there’s no distinction between two types of crossings.
And now we come back to eversions. We write the sphere as a 2-dimensional cup followed by a 2-dimensional cap. Since we have duals, we can consider one side to be “painted red” and one side “painted blue”. One way of writing the sphere has the outside painted red and the other side is painted blue. An eversion in our language will be an explicit list of 3-morphisms that run from one of these spheres to the other.
Scott Carter and Sarah Gelsinger have now created just such an explicit list of directions to evert a sphere. And, what’s more, they’ve rendered pictures of it! Here, for the first time in public, is a 50MB PDF file showing the Carter-Gelsinger eversion.
First they illustrate the basic pieces of a diagram of knotted surfaces (pp. 1-4). Then they illustrate the basic 2-morphisms that build up surfaces (pp. 5-6), and write out a torus as an example (p. 7). Then come a few more basic 2-morphisms that involve self-intersections (pp. 8-9) and a more complicated immersed sphere (pp. 10-11). Each of these is written out also as a “movie” of self-intersecting loops in the plane. Next come the “movie moves” — the 3-morphisms connecting the 2-morphism “movies” (pp. 12-17). These are the basic pieces that let us move from one immersed surface to another.
Finally, the eversion itself, consisting of the next 79 pages. Each one consists of an immersed sphere, rendered in a number of different ways. On the left is a movie of immersed plane curves. On the top are three views of the sphere as a whole — a “solid” view on the right, a sketch of the double-point curves in the middle, and a “see-through” view on the left. The largest picture on each page is a more schematic view I don’t want to say too much about.
The important thing to see here is that between each two frames of this movie is exactly one movie move. Everything here is rendered into pictures, but we could write out the movie on each page as a sequence of 2-morphisms form the top of the page to the bottom. Then moving from one page to the next we trace out a sequence of 3-morphisms, writing out the eversion explicitly in terms of the basic 3-morphisms. As an added bonus, there’s only ever one quadruple point — where we pass from Red 26 to Blue 53 — and no higher degree points.
I’d like to thank Scott for not only finishing off this rendering he’s been promising for ages, but for allowing me to host its premiere weblog appearance. I, for one, am looking forward to the book, although I’m not sure this one will be better than the movie.
[UPDATE] Some people have been having trouble with the whole 50MB PDF (and more people might as the Carnival comes to see this page. Scott Carter broke the file up into five pieces, and I’ve put them up here in a new post.
There’s a glitch in part 4, but I’ll have that one up as soon as I can.