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January 4, 2017

Globular for Higher-Dimensional Knottings (Part 3)

Posted by John Baez

guest post by Scott Carter

This is my 3rd post a Jamie Vicary’s program Globular. And here I want to give you an exercise in manipulating a sphere in 4-dimensional space until it is demonstrably unknotted. But first I’ll need to remind you a lot about knotting phenomena. By the way, I lied. In the previous post, I said that the next one would be about braiding. I will write the surface braid post soon, but first I want to give you a fun exercise.

This post, then, will describe a 2-sphere embedded in 4-space, and we’ll learn to try and unknot it.

Loops of string can be knotted in 3-dimensional space. For example, go out to your tool shed and get out your orange heavy-duty 25 foot long extension cord. Plug the male end into the female and tape them together so that the plug cannot become undone. I would wager that as you try to unravel this on your living room floor, or your front lawn, you’ll discover that it is knotted.

Rather than using a physical model such as an extension cord, we can also create knots using the classical knot template of which I wrote in the first post. There you create knots by beginning with as many cups as you like in whatever nesting pattern that you like. For example:

And yes, these nestings are associated to elements in the Temperley-Lieb algebra. Then you can click and swipe left or right at the top endpoints and thereby entangle strings as you choose:

Close the result with a collection of caps, and a link results:

It is possible that the resulting link can be disentangled. To play with Globular, keep your link in the workspace, click on the identity menu item on the right, and then start trying to apply Reidemeister moves to it. For example, when I finished simplifying my diagram, I got the trefoil. I didn’t expect this!

If you want to see how I got the trefoil, you can look at my sequence of isotopy moves within globular.

By clicking the identity button on the right, you preserved the moves you used. The graphic immediately above indicates an annulus embedded in 3-space times an interval [0,1][0,1]. At the bottom is the knot that I drew, at the top is the result of the isotopy.

You can go into that isotopy, click the identity button, and modify it further to find a more efficient path between the knots!

Just as circles can be linked and knotted in 3-space, surfaces can be knotted and linked in 4-space. The knotting of higher dimensional spheres was observed by Emil Artin in a 1925 paper. Most progress about higher-dimensional knots occurred in the era circa 1960 through 1975. At that time, new algebraic topological techniques, particularly homological studies of covering spaces, occurred. Some authors, Yajima in particular, also initiated a diagrammatic theory. The diagram of a knotted surface is its projection from 4-space into 3-space with crossing information indicated. I like to think of the diagram as representing the knotted surface in a thin neighborhood of 3-space. The bits of surface that are indicated by breaks protrude into 4-space in that thin neighborhood. Still this imagery does not help manipulate the surface. By analogy, if you think of a classical knot as being confined to a thin sheet of space, then you’ll feel constrained in pulling the under-crossing arc.

As sighted humans, we perceive only surface. We posit solid. So as I sit at my desk, I see its top and I presume that it is made of a thick wood. The drawer in front of me defines a cavity in which paper clips, rubber bands, and old papers sit. But I can’t see through this. I only see the front of the drawer. When I look at the diagram of a knotted surface, I create visual tropes to help me understand. How many layers are there behind the visible layer? Where does the surface fold? Where does it interweave? Within the globular view (project 2) of a knotted surface, we see (1) the face of the surface that lies closest to us, and (2) the collection of double curves, triple points, folds, and cusps that induce the knotting. Globular is new — only a year old. So its depiction of these things is not as elegant as it might be, but all the information is there. Mouse-overs let us know the type and the levels of all the singular sets. Cusps and optimal points of double curves (these are double curves in the projection of the surface into 3-space not double curves in 4-space) have the same shape. They should have different colors. Similarly birth, deaths, saddles, and crotches will all be cup or cap like. Hover the mouse to the critical point, and you’ll see what it is.


is the image of a sphere in 4-space that looks like it might be knotted. But in fact it is not. This is the image from a worksheet that I created specifically for the energetic readers of this blog. In the worksheet, I created a sphere embedded in 4-space that is constructed as Zeeman’s 1-twist spin of the figure-8 knot (4sub1) in the tables. At least I think I did! Zeeman’s general twist spinning theorem says that the n-twist-spin of a classical knot is fibered with its fibre being the (punctured) n-fold branched cover of the 3-sphere branched along the given knot. When n=1, this branched cover is the 3-ball, and so the embedded sphere bounds a ball, and therefore is unknotted.

The worksheet that I created here is a quebra-cabeça — a mind-bending puzzle for the reader. Can you use globular to unknot this embedded sphere? By the way, I am not 100 percent sure that I constructed this example correctly ;-) But here is my advise for unknotting it. There are two critical points, one saddle and one crotch, that need to have their heights interchanged. To interchange these heights, add two swallow tails: a left (up or down) swallow tail (L ST (up or down)) on the interior red fold line, and a right (down or up) (R ST (down or up)) on the interior green fold. These folds are mouse-over named cap and cup, respectively. The swallow tails allow you to turn the surface on its side. Then pull the stuff (type I, ysp, and psy) that lie along these folds into the swallowtail regions. meanwhile interchange the heights of the crotch and saddle. When you get done with that, I’ll give another hint, and I may have done these operations myself.

Posted at January 4, 2017 6:08 AM UTC

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Re: Globular for Higher-Dimensional Knottings (Part 3)

I’ve been tinkering with it for quite a while but then I tried a move which had worked before but it just crashed globular instead. I only wish I knew what went wrong there.
Anyway, assuming I interpret the diagram right (which is not at all a given), I’m struggling to invert the shape. What I’m trying to do is to pull the two swallow tails all the way out, but I just end up getting stuck at some point. I’m not quite sure yet what some of the moves actually mean. Some of the ones I don’t quite get are apparently ones that are implemented by default: I can’t see you having defined those anywhere. So I can only assume they are Reidemeister moves. (I’m not very familiar with knot theory) Some of those predefined moves seem to be unstable and occasionally crash globular though.
I wish there was a 3D slice view alongside the 2D slices. Occasionally I’m kind of lost as to why some move (especially if it’s about pulling wires over each other) doesn’t work and I have to do a quite different move which is called “pullover interchanger above/below” or something like that and only then do the wires cross over each other.
I assume that’s because I’m not actually looking at wires but rather at surfaces that are just cut through, but I have a hard time visualizing what’s going on.

Posted by: kram1032 on January 4, 2017 1:50 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

I know of some small problems with certain canned moves. Also know that within my movie move template(1612.004v1), the move “Arc thru R cusp -=>+” has an error in it. The latter is less fatal since it can be recovered from a pre-processed move. Or if you look at (Arc thru R cusp +=>-) you’ll see that it is the same as (Arc thru cusp -=>+) so you can fix this by creating a new move. On the other hand, I will fix this soon so that error may not persist for long.

For the more serious bug, within 1607.003v1, I defined a number of moves, such as (TIII cup -++). I think that there are 12 variations cup<->cap, and +++,++-,+–,-++,–+,—. Each of these moves is a consequence of a move that may cause a crash. The crash is that you perform the move, and then the program freezes. You can go back one step, hit cntrl-m, and do something else, but if you need the move, you can add my 12 moves. I have proofs that they are equivalent to the moves that crash or their inverses. So since my moves, or their inverses hold, you get the move you need.

In general when you get the crash, try to locate it and send a crash report.

Meanwhile, your description of what to do with swallow tails is not what I had in mind for the exercise.

Posted by: Scott Carter on January 4, 2017 8:28 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

Hi, thanks a lot for the bug report. If you can reliably reproduce it, can you export the workspace just before triggering the bug, and then upload that to the globular issue tracker? Thanks!

I agree it can be hard to see what’s going on when you’re slicing through a high-dimensional proof. In the future there will be a 3d display that will help somewhat, I hope.

Posted by: Jamie Vicary on January 4, 2017 10:15 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

This is the bug I told you about last July. The steps to reproduce are as follows: (directly quoted from nForum thread)
1607.003v1; open Prop 4; go to source and hit r i (restrict, identity);
then apply def ysp up at 15; def ysp up inv at 19; pull-through right tangle interchanger above inverse at 20; and pull the pull-throughs to either side of the interchanger between them so they are at 16 and 18. Then try to apply Int-R-S inverse. This will fail and quickly lead to Globular becoming unusable.

Also, pull-through pull-through can cause trouble:
1606.002v1; open Lemma_proof and go to target (and restrict);
pull down the swallowtail_ev_x as far as possible; then pull the interchanger at 7 up and select choice 1. This will lead to a crash if you attempt to work much further.

Posted by: Eyal Minsky-Fenick on January 4, 2017 10:51 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

Dear Eyal,

If you go into the proof of Prop 4, you’ll see the work around that I found for this bug. I admit that it is not very elegant! Now I pretty much understand what the bug is (and under some circumstances, I am not convinced that it is a bug) and within the movie move context, I can put back in the TIII cup/cap (sign, sign, sign) moves which will be a temporary fix. Also, I will follow Jamie’s advice, and send in a bug report when I put those moves in (since there, the bug is more isolated). I hope to complete that before Jan 6, 2017 4:00PM CDT. I imagine that I will do so more quickly since doing so is more fun than preparing syllabi.

Posted by: Scott Carter on January 5, 2017 3:56 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

I have not finished the revisions. I’ll get them in this weekend or on Monday morning. I did include some of the TIII cap moves.

Posted by: Scott Carter on January 6, 2017 9:53 PM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

Speaking of the issue tracker. May I suggest adding a tab to the Globular interface called “Feedback”? It would link right to where feedback of that kind is supposed to be given.

I largely like Globular’s slick and minimal interface but in some instances it’s almost a bit too minimal.
Also I think you must have some serious inefficiencies when it comes to renaming stuff. Just clicking into text fields after you did some proof work can suddenly take ages. It can be kind of painful.

Posted by: kram1032 on January 7, 2017 2:04 AM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

Yes, sorry, I meant to do this this afternoon, but I got confused by the github interface. I can probably implement several different instances of the error. I’ll check carefully on the next run throughs.

Posted by: Scott Carter on January 7, 2017 3:35 AM | Permalink | Reply to this

Re: Globular for Higher-Dimensional Knottings (Part 3)

I finally finished the revision of the movie move template to include the moves that are dependent upon the IntI0 moves. The result is here.

Soon I hope to have the 4th and 5th posts to this series.

Posted by: Scott Carter on January 12, 2017 4:56 PM | Permalink | Reply to this

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