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Stream: theory: category theory

Topic: periodic table of string diagrams

Simon Burton (Nov 04 2025 at 14:32):

I'm thinking about a "periodic table of string diagrams" that is to do with diagrams of n-manifolds inside of an (n+k)-dimensional manifold (probably a sphere). It starts getting tricky with 2-manifolds inside 2+k dimensional spheres. Although 2+1 seems not too tricky, and 2+2 is where knotted surfaces live, but where does it "stabilize" ? Is it in 6=2+4 dimensions? And 5 dimensions have some kind of "sylleptic" surface knots? What are these things?

John Baez (Nov 04 2025 at 15:47):

As you probably know, this issue is the topic of the Tangle Hypothesis that James Dolan and I promulgated here.

The study of embeddings of n-manifolds in (n+k)-manifolds stabilizes when k = n + 2, just as the periodic table of n-categories predicts.

John Baez (Nov 04 2025 at 15:49):

Sylleptic monoidal bicategories should be important for 2-manifolds knotted in 2+2+1 dimensions, just as braided monoidal categories are important for 1-manifolds knotted in 1+1+1 dimensions.

The study of knotted 2-manifolds stabilizes, and becomes boring, in 2+2+2 dimensions, just as the study of knotted 1-manifolds stabilizes in 1+1+2 dimensions.

John Baez (Nov 04 2025 at 15:49):

If we were talking in person I could argue for this by waving my hands around in 5 dimensions.

John Baez (Nov 04 2025 at 15:53):

In Cambridge, one of those students of Jamie you met is working on sylleptic monoidal bicategories. I tried to persuade him to look for examples to construct invariants of knotted 2-manifolds in 5-dimensional space.

Simon Burton (Nov 04 2025 at 16:04):

image.png
Aha, thankyou, this seems to be the periodic table i was wondering about, from your paper with James Dolan.

Simon Burton (Nov 04 2025 at 16:08):

image.png
And here is the stable ("abelian") region

Simon Burton (Nov 04 2025 at 16:10):

I also found this mathoverflow answer discussing linked surfaces in 5d.

John Baez (Nov 04 2025 at 19:16):

That's a nice answer. I've never understood the "metastable range" of homotopy theory, which comes before stabilization kicks in.

Simon Burton (Nov 06 2025 at 13:31):

I've realized theres a glitch here. The tangles, which are pictures of globular morphisms, are poincare dual to string diagrams. I'm hoping/guessing that the tangle framing is enough structure that this is not an issue. Anyway, here is a first attempt at a Poincare dual table of string diagrams:
image.png

John Baez (Nov 06 2025 at 13:32):

No, the tangles are the string diagrams - they are not Poincare dual.

Simon Burton (Nov 06 2025 at 13:34):

Looking at these two periodic tables it seems like you are right! The dimensions are actually agreeing :-)
I think I got confused by the letters (as usual).

Jules Hedges (Nov 28 2025 at 12:32):

Simon Burton said:

I'm thinking about a "periodic table of string diagrams" that is to do with diagrams of n-manifolds inside of an (n+k)-dimensional manifold (probably a sphere)

How common is the idea that (higher) string diagrams live on a sphere? I generally think of them as inside an $n$-dimensional ball or cube (depending if you are doing globular or cubical things). I've come across the idea that 2-dimensional string diagrams live not inside a square but inside $\mathbb P \mathbb R^2$ with the top and bottom boundary collapsed to a "virtual node" at infinity, which you can also do on the surface of a sphere, is that what you have in mind?

Amar Hadzihasanovic (Nov 28 2025 at 12:41):

Well, it is helpful to think for example of a planar graph as being embedded onto the surface of a sphere, since then up to the appropriate notion of isotopy the graph is described by the data of a spherical polytope, whose dual is also a spherical polytope, which determines the dual planar graph.

Amar Hadzihasanovic (Nov 28 2025 at 12:45):

But yes, to actually recover the embedding into the plane you have to also specify one face of the spherical polytope, which becomes the "outer face", which when removed produces a 2-dimensional ball.

Amar Hadzihasanovic (Nov 28 2025 at 12:46):

I'd say it's useful to keep both perspectives in mind.

John Baez (Nov 28 2025 at 13:22):

Jules Hedges said:

How common is the idea that (higher) string diagrams live on a sphere? I generally think of them as inside an $n$-dimensional ball or cube (depending if you are doing globular or cubical things).

That's more typical. Thinking of these diagrams as living on a sphere imposes new relations, since you can move a string around the back of the sphere - but these extra relations are interesting and important at times, especially in quantum topology. That's why people study spherical categories and spherical 2-categories.

David Corfield (Nov 28 2025 at 14:22):

There was an approach to mediating between cubical and spherical settings discussed by Noah Snyder at the Secret Blogging Seminar in a post The operadic periodic table:

image.png

The group that ran that blog were Berkeley PhDs working in quantum topology, fusion categories and the like.

Jules Hedges (Nov 30 2025 at 15:41):

Oh, that's interesting, haven't seen that "symmetry" axis before

Jules Hedges (Nov 30 2025 at 15:43):

I can't see how to extend the pattern though... what's the rule that goes along the top row like $I^3, I \times S_1, S_2$?

Jules Hedges (Nov 30 2025 at 15:47):

Aha maybe I should write $B_n$ for an $n$-dimensional ball and then see it as $I^2 \times B_1, I \times B_2, B_3$ ?