Post #2516083
2026-04-17 15:59 UTC
One version of our main theorem can be explained in Pₓ, the free symmetric monoidal category on an invertible object x. It says that morphisms in Pₓ are characterized by the *parity* of how many instances of 8ₓ they have.
In particular, the composite or sum of two figure eights is the identity! Our fantastic choice of notation expresses this fact as follows:
8ₓ∘8ₓ = 8ₓ+8ₓ = 1₀.
So, this fact implies that any composite or sum of figure eights can be reduced to just *odd* or *even*. The main theorem says, moreover, that *every* morphism in Pₓ boils down to some composite or sum of figure eights. A little later in this thread I'll give some more precise (more comprehensible) versions of the same result.
[Aside: I want to pause and note that this might sound familiar to some readers, because these facts have been known in some form or other for a *long time*. They are very well studied! Our paper has a "Relation to Literature" subsection that addresses some of this, and I'll make some further comments below, but this thread is mostly for people who haven't seen it before, or have seen it but would like to see an alternative explanation because it's neat.]
(3/9)
Replies (1)
-
@nilesjohnson@mathstodon.xyz 2026-04-17 16:00
Before that, here's an example of two other composites that might appear different but are actually equal to each other and to the figure eight. We call one Cₓ, the *figure C*, and the other Hₓ, the *figure H*, since the string diagrams sort of look like those letters. So, Cₓ and Hₓ also have *odd* parity, because they're each equal to one instance of 8ₓ. Not pictured: There is also a "reverse C" that uses the braiding of x' with itself, and a figure eight on x' that reverses the roles of x and x'. Both of these are also equal to 8ₓ = Cₓ = Hₓ, and therefore have odd parity. (4/9)