Higher-dimensional commutative laws

If you click on the link, you'll see '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
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