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I wish this article talked more about how this plays out in the 3-dimensional analogue (knot theory) for comparison. I have very little picture of how I would
by Sniffnoy 2y ago
I wish this article talked more about how this plays out in the 3-dimensional analogue (knot theory) for comparison. I have very little picture of how I would expect this to work, so it would be helpful to start with a case I can picture, and then say how this is similar and how it's different.
- VyseofArcadia 2y agoI wouldn't call knot theory the three dimensional analog. They spend a lot of time talking about analogs you can visualize, like the sphere and the torus.
- Sniffnoy 2y agoWell, it's discussing having some larger ambient manifold and then embedding other manifolds in it of codimension 2 (and then also taking the complement and studying the fundamental group of said complement). If the ambient manifold has dimension 3 instead of 4, that's knot theory. Perhaps there's some reason why that's not the appropriate analogue, but it's certainly the naive analogue; if there's some reason it's not actually the right analogue, you're going to have to explain it. And, again, the point isn't just to have an analogue that one can visualize, but a simpler case where the results are known so that one can compare the results and see how the dimension affects things.
- deleted 2y ago[deleted]