**Higher-dimensional commutative laws**

If you click on the link, you'll see +Scott Carter's picture of the two sides of the

**Zamolodchikov tetrahedron equation**, a tongue-twisting and brain-bending equation that shows up in topology.

My blog article explains it, with pictures. But in simple terms, the idea is this. When you think of the commutative law

xy = yx

as a

*process*rather than an equation, it's the process of switching two things: in this case, the letters x and y. You can

*draw*this process using two strings that switch places: that is, a very simple "braid" like this:

\ /

/

/ \

It turns out that this braid obeys an equation of its own, the

**Yang-Baxter equation**. This is easy to explain with pictures, but it's hard to draw pictures here, so visit my blog article.

If you then think of the Yang-Baxter equation as a process of its own,

*that*process satisfies an equation: the Zamolodchikov tetrahedron equation. This equation really wants to be drawn in 4 dimensions, but you can get away with drawing it in 3 - just as I drew that simple braid on the plane.

This goes on forever: whenever you reinterpret a equation as a process, that process can (and usually should) obey new equations of its own. As you do this, you naturally go to higher dimensions. The Zamolodchikov tetrahedron equation is a nice example of how this works!

#topology #4d

- I am going to add some commentary and links here. First, this illustration is one of 8. Seven of these are complete and are found in the beamer presentation here: http://www.southalabama.edu/mathstat/personal_pages/carter/FrohmanBirthday2016JSChandout.pdf

I haven't had a chance to finish drawing the eighth move.

These 8 moves simultaneously relate higher associativity and commutativity.

On page 35 of this preprint http://www.southalabama.edu/mathstat/personal_pages/carter/FOAM20150914.pdf you'll see the next higher relation. In addition, this paper contains the 16 moves of movie moves that correspond to the higher relations for commutativity and associativity.

The original 8 moves are analogues of some of the Roseman moves for knotted foams in 4-dimensional space.

In my paper on Reidemeister/Roseman moves for foams, there are two moves (at least) that I didn't take into consideration. The illustration that John shows here is one that will appear in that paper (or some version thereof).

Behind all of this work, there is a bit of higher category theory. I will discuss a little of this in the next comment.48w - As promised: Start from a category whose objects are the non-negative integers represented as dots on a line in unary notation. Consider morphisms X, \overline{X}, \cap, \cup, Y, and \lambda (draw as an upsidedown Y). Also include I as the identity.

Mimic the category of tangles. Let families of dots transform via vertical compositions of Xs Ys and so forth. And let the tensor product be juxtaposition. Don't impose relations, but be cautious about the tensor product of morphisms (they shouldn't appear at the same time level). Now all of the relations that you'd like to be true are 2-morphisms. For example page 23-29 of http://www.southalabama.edu/mathstat/personal_pages/carter/FrohmanBirthday2016JSChandout.pdf

Next you can assert relation among these. Those are the Roseman-type moves. Or you can keep assuming that relations only hold up to the next order equality.

The measurements of these phenomena can be achieve via certain cocycles that are analogous to group cocycles. Sometimes they are group cocycles.48w - By the way, the
**third Reidemester move**seems to also be a reversal of the orginal braid, doesn't it? Please tell me why. I am just a high school student. I want to understand.48w - +Vedanth Bhatnagar - you're right that the 3rd Reidemeister move goes from the braid "121" to the braid "212", where "121" means that first strands 1 and 2 cross, then strands 2 and 3, then 1 and 2... while "212" means that first strands 2 and 3 cross, then 1 and 2, then 2 and 3.

However, to understand the pattern that begins here:

commutative law

third Reidemeister move

Zamolodchikov tetrahedron equation

I find it useful to think of the third Reidemeister move in a different way, which is explained in my blog article: it's "sliding a crossing over a crossing". Then the Zamolodchikov tetrahedron equation is about "sliding the process of sliding a crossing over a crossing over a crossing"! And it keeps going...48w

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