## July 7, 2008

### Sphere Eversion

#### Posted by David Corfield

guest post by Scott Carter, a commentary on his and Sarah Gelsinger’s sphere eversion (50MB). See also John Armstrong’s commentary.

Consider the $2$-category given as follows. The objects correspond to a (possibly empty) set of dots along a vertical axis. Each dot has an associated sign: $-$ will indicate that a $1$-morphism emanating from the dot should be a left pointing strand; $+$ indicates a right pointing strand. The orientation information will be completely suppressed here. The $1$-morphisms are generated (in the sense of composition and tensor products) by $\subset$, $\supset$, and $X$. Of course a predominantly horizontal arc indicates the identity $1$-morphism. There are two flavors of $\subset$ and $\supset$ when orientations are drawn, and 4 flavors of $X$ in that case.

A $1$-morphism from the empty object to the empty object is a collection of generically immersed circles in the plane.

The set of $2$-morphisms are generated by birth, death, saddle, cusp,(the projection of) type II and type III, and the (projection of) Yetter $\psi$ move in which a double point on one arc near an optimum bounces to the other arc near the optimum. Here, of course, height is measured in a left to right direction. Finally, there are tensorators which allow the interchange of critical levels from right to left.

We want a strict symmetric monoidal $2$-category with duals on an object generator $(*)$ whose dual is $(-*)$. So we should define those identities among $2$-morphisms that are allowed. Notice first though that there is no pivotal axiom that would correspond to the type I move. Each of type II, type III and the $\psi$ move are isomorphisms. Loosely speaking their squares are the identity. More precisely, we can assume a naturality axiom for the transposition $X$. And one case of naturality is the $\psi$ move. The naturality axiom, then, is among the $2$-morphisms. There is a direction associated to the move, and the move followed by its opposite is the identity. Similarly there is a direction for type II. Both compositions of type II are equivalent to the identity $2$-morphism. The critical cancelation between a saddle and an optimum (birth or death) is axiomatized as in Baez-Langford. The Zamolochikov relation holds, and a slew of relationships between folds, optima of fold sets, cusps, and double and/or triple points hold. I think that Baez-Langford arguments give that this immersed surface category is the FREE SYMMETRIC MONOIDAL $2$-CATEGORY WITH DUALS ON AN OBJECT GENERATOR.

All of the generating $2$-morphisms (with the exception of the tensorator) are illustrated in the file. Next the $3$-morphisms (or identities among $2$-morphisms) are shown.

A red embedded sphere represents a particular $2$-morphism from the identity $1$-morphism on the empty object to itself. Each step in the eversion represents a relationship among $2$-morphisms. The entire eversion can be thought of as a $3$-morphism from the red $2$-morphism to the blue $2$-morphism.

Smale’s theorem says that there is such a $3$-morphism in this category. It is obvious (or an easy exercise with orange peels) that there is such a $3$-morphism in the pivotal case.

Our eversion differs from all of its predecessors in that every step is a generic codimension $1$ singularity. Consequently, it is not symmetric with respect to the quadruple point.

Scott

Posted at July 7, 2008 9:07 AM UTC

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### Re: Sphere Eversion

I hadn’t known of the mysteries of sphere eversion before. Watching the video was a surprising amount of fun.

Now I am trying to see what is going on here. On the machine in my current office any attempt to download the 50MB file linked to above fails. I am lacking any information that might be hidden there.

So let me just try to understand what the entry here is saying:

Consider the 2-category given as follows. The objects correspond to a (possibly empty) set of dots along a vertical axis. Each dot has an associated sign: − will indicate that a 1-morphism emanating from the dot should be a left pointing strand; + indicates a right pointing strand. The orientation information will be completely suppressed here. The 1-morphisms are generated (in the sense of composition and tensor products) by $\subset$, $\supset$, and $X$.

So we have a bunch of arrows drawn on the line, touching each other at their endpoints (but not necessarily target-source but also source-source and target-target), and we want to generate a monoidal category from them, I suppose.

Do I read the above as saying that $\subset$ is to denote concatenation of arrows, while $\supset$ is to denote their monoidal tensor product? Why, then, that unusual notation. And what is $X$? Or is that a typo for $\times$?

I am afraid I didn’t understand the message of this first paragraph. Can anyone help me?

Posted by: Urs Schreiber on July 7, 2008 4:02 PM | Permalink | Reply to this

### Re: Sphere Eversion

Hi Urs,

I read it slightly differently then you. The objects are either lines with a bunch of +/- labeled points of equivalently a bunch of in/out arrows.

The elbow and coelbow (I don’t know exactly how to make those symbols) are morphisms going from a plus point and a minus point to no points (or the opposite). So roughly these are the morphisms which say the + point and - point are dual. The tensor product is stacking lines, and the (symmetric) braiding of just two points is the X. That is why there are four kinds of Xs (++) (+-) (-+) (–).

Does that help?

Posted by: Chris Schommer-Pries on July 7, 2008 4:51 PM | Permalink | Reply to this

### Re: Sphere Eversion

For elbow/coelbow I was just using \subset and \supset.

Posted by: David Corfield on July 8, 2008 1:13 PM | Permalink | Reply to this

### Re: Sphere Eversion

$\subset$ $\supset$ Okay. Got it!
Posted by: Chris Schmmer-Pries on July 8, 2008 1:59 PM | Permalink | Reply to this

### Re: Sphere Eversion

I really am sorry about the file size!

Chris has the definitions correct. I am mimicing the category of tangles, but using
left/right instead of up down. The zig-zag move that ordinarily defines a duality (or the existence of a non-degenerate bilinear form in the vector space case) is now a 2-isom. Schematically, the 2-isom is a cusp:
Think of the surface (x, x^3 - a x , a) for
x between -1 and 1 and a between -1 and 1.
That it is a 2-isom are the lips and the beak-to-beak moves.
The vertical lines on which the objects live are parallel, and a 1-morphism is an immersed curve in the plane with a non-degenerate critical points at distinct critical levels.

Posted by: Scott Carter on July 7, 2008 6:16 PM | Permalink | Reply to this

### Re: Sphere Eversion

Thanks Chris, thanks Scott. Now I see what is meant. How does that differ from the 2-category you (Chris) describe by generators and relations, if at all?

Still enthusiastic due to the cool video visualizing sphere eversion, now I am really interested in this statement:

Our eversion differs from all of its predecessors in that every step is a generic codimension 1 singularity.

My Firefox still gives up when trying to open the 50MB file….

Posted by: Urs Schreiber on July 7, 2008 7:05 PM | Permalink | Reply to this

### Re: Sphere Eversion

I will have to repackage the figures into several files. Arrrgh!

If you look at Baez–Langford or C–Rieger–Saito or the C–Saito book, then you can guess which movie moves are needed to evert: those that do not involve branch points. Each movie move represents a certain codimension 1 singularity in the space of maps between surfaces: yes surfaces. When the sphere is immersed in 4-space you look at it. When you look the sphere is projected to the retina. When the sphere is illustrated it is projected onto the plane of the page.

Outside-in is highly symmetric, and its singular point has many sheets converge. Ours only involves an isolated and a generic quadruple point.

Posted by: Scott Carter on July 7, 2008 9:45 PM | Permalink | Reply to this

### Re: Sphere Eversion

I inserted a link to the Baez–Langford paper. Why don’t you put the Carter–Rieger–Saito paper on the arXiv? It’s important… and it has a lot of pictures needed to understand our paper!

Posted by: John Baez on July 12, 2008 8:22 AM | Permalink | Reply to this

### Re: Sphere Eversion

That is a good idea.
I will mention it to Masahico. I haven’t heard from Joachim in years. I am not sure if we have an electronic copy somewhere.

In a certain sense, by redrawing the non-branch point
movie moves, I was addressing that need. On the other hand there is all the X cap cup code in there. In LLT
there is an alternative coding that was useful for computer implementation.

Posted by: Scott Carter on July 14, 2008 2:06 AM | Permalink | Reply to this

### Re: Sphere Eversion

Urs wrote:

My Firefox still gives up when trying to open the 50MB file.

Did you try downloading it to a file then opening it with Adobe Reader (or whatever), rather than trying to view it directly within Firefox? I can imagine Firefox’s PDF-viewer plug-in gagging on a 50Mb file, but Firefox really shouldn’t care about the file size when you simply download to disk.

Posted by: Greg Egan on July 9, 2008 5:39 AM | Permalink | Reply to this

### Re: Sphere Eversion

This is really impressive!

Posted by: Bruce Bartlett on July 8, 2008 10:33 PM | Permalink | Reply to this

### 3-d graphics

On a related note, does anyone have any good ideas for getting 3-d graphics on the web?

I tend to use maple to do a lot of visualation things: curves and surfaces in 3d, often animated. That’s good because I can code up animations in maple very quickly. The problem comes when I want to show them to other people. I can’t find a sensible way to export such graphics from maple and get them viewable on the web.

Here’s a very low-bandwidth (4KB) animated gif showing the trajectory of some pairs of particles in the plane being created and anihilated, so that the trajectory forms a trefoil.

The full version in maple looks a lot more impressive and can be rotated by hand as the animation is running.

Animations can be exported as animated gifs, but these come out very large and have to be compressed in some other software, it would seem more sensible to be able to save as some sort of mpeg file, or possibly something else more appropriate.

As for 3-d things, well one possibility is to do an animation of the view rotating and save it as an animated gif, but that is not very satisfactory. Another possibility is to try to export as a vrml file, but I have had absolutely no success with this – trying to view the resulting files with the cortona browser plug-in gave no joy – I get the impression that this is not well supported. The other options for exporting are POV and DXF, but I have no idea how to put those up on the web.

So any suggestions? Can I do anything sensible with maple or do I have to learn something completely new like flash? How do you do your pictures Scott? Greg?

Posted by: Simon Willerton on July 8, 2008 11:52 AM | Permalink | Reply to this

### Re: 3-d graphics

I drew my stuff with Illustrator, using a mouse. Kenny Baker may be the person to ask. It is a bit strange, though, that vector based graphics of Bezier type curves get to be so large so fast. I guess that the youtube videos are pretty 1-dimensional ;-)

Posted by: Scott Carter on July 8, 2008 2:59 PM | Permalink | Reply to this

### Re: 3-d graphics

Simon, for fixed illustrations I just export animated GIFs from Mathematica. For anything that needs user interaction I use Java.

I’ve never really looked into Flash, so I don’t know much about its capacities, though I’d imagine it’s the most widely supported format.

Posted by: Greg Egan on July 9, 2008 12:09 AM | Permalink | Reply to this

### Re: Sphere Eversion

I cut the file into smaller pieces, but made a mistake on the 4th piece. I will fix that tonight. Meanwhile, smaller files are found at Unapologetic.

Posted by: Scott Carter on July 10, 2008 7:02 PM | Permalink | Reply to this

### Re: Sphere Eversion

The repaired part 4 is available now. Have at it!

Posted by: John Armstrong on July 11, 2008 1:43 AM | Permalink | Reply to this

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