The n-Category Café

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 $[0,1]$. 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.

Here:

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.