Using Globular to isotope and create higher dimensional knottings.

1. Classical Knots

About 7 months ago, Jamie Vicary contacted me with a globular worksheet of which, initially, I could neither make heads nor tails. He patiently explained to me that what I was looking at was an example that I had worked out for Bruce Bartlett one evening and how to read it. Fast forward through a number of late night (for him) skype sessions and a number of heartbreaking system errors for me, and now I feel that Globular is not only the best way to do higher dimensional knot theory and diagrammatic calculations, but it has the potential to be revolutionary. It will give insight into classical theorems and it will be used in the near future to create diagrammatic proofs of new theorems.

The papers to read about globular are Data structures for quasistrict higher categories by Jamie Vicary and Krzysztof Bar, and Globular: an online proof assistant for higher-dimensional rewriting in which Aleks Kissinger joins Jamie and Krzyztof to explain further and give some nice examples of globular's potential.

This is the first in a series of blog entries about globular and how to use it. Specifically, I am framing these worksheets with exercises. I have spoken about them here, here, and here. There are not recordings of those talks, so I want to document some of that work for others. The blog entries for the café are split into a discussion of a number of worksheets. These are, perhaps, 2.0 versions of standard worksheets to do some diagrammatic calculations from my own point of view. Here you should learn how to view and work with them. But before that, you should know that there is still room for improvement.

Most importantly, these worksheets are for unoriented knots and a knotted surfaces. Indeed, the hope here is to be able to address some key questions about non-orientable surfaces. If you want to create an oriented theory, then you need to add more objects and relations among them. I may create some of these soon. There will be differing color conventions within those worksheets. Color choices are helpful to the user to inform user of what will happen. Also, I am interested in knotted trivalent graphs and knotted foams. I have not yet created templates for their manipulations. To return to an old hobby, globular is a lego set made of small pieces that you can create on your own, and here I am saying, "Hey, mommy look what I have done so far."

The first worksheet to view is the classical knot template. Here we start in a 2-dimensional black ambient space. The first object that we see is a white dot. Next we create a cap and a cup. The cap is, from my point of view, a $1$-morphism with source a pair of points and target the empty object. A cup has source and target reversed. The Reidemeister type II and type III moves are intrinsic to globular, but the type I move has to be added. There are 6 variations of type I moves, and only two need to be added. In this worksheet, I have shown how two of the others follow from these two and the psy/ysp moves. The ysp moves, in which goes to , are consequences of the categorical axioms. As an exercise, try and figure out what the remaining two type I moves are, and give proofs using globular.

Also, zig-zag identities are not intrinsic. So left zig-zags and right zig-zags, and their inverses have been added.

There is a learning curve to using globular. You probably need a 3-button mouse and google chrome to maximize the experience. Within this template first click "allow undo" so this is turned off. Next left click on cell 2. A white dot appear. Click again and attach a second and third dot. Whether or not you attach above or below does not matter. Now click the identity button on the right. Move your mouse to the project box and change project to 1. Project is the number of dimensions you are suppressing. So at project 1 you are looking at a 2-dimensional projection of three straight arcs in 3-dimensional space. Let us braid as if they are locks of your little sister's hair. Run your mouse to the bottom of the middle of the middle strand. Click left and swipe left. Go back to the middle; click left and swipe right. Continue at will. Now move your mouse pointer into the middle of one of the arcs. My favorite locale is at one of the inflection points. Click left and swipe left or right. Do this a few times, and see if you can create triangles at which the Reidemeister type III move (YBE) can occur. To affect a Reidemeister type III move select (left-click) either the top-most or lower-most crossing among the three that define a potential move and that is either a top/middle or bottom/middle crossing. Then swipe the mouse in the direction of the move as if you are pinching the crossing between your fingers (cat's cradle) and moving it.

As a final exercise with this template, go to Knot info, for example, and figure out how to construct the braid closer of each of the knots with fewer than, say 11 crossings. After having constructed a braid word (for example (1,1,1), hover the mouse to the left or right until you see the mouse-over say cell 1 @ [?,?], then left click to tensor with an identity string. Do so for as many strings as needed (in this case 2). Then left-click at the top/bottom middle string at which you want to create a cup or cap. A menu will pop-up with choices (in my case it says 1. cap 2. cap). Hover in the menu until a yellow rectangle shows up. The rectangle indicates where the cap will be attached. Do so until you have constructed the braid closer. Then use the classical knot template to demonstrate that each braid closure can be isotoped into its more standard view.