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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).