Daniel Tubbenhauer on Fast, Strong Knot Invariants

My comment:

These are fun talks, thanks for sharing them. One of the reasons I am interested in smooth manifolds and Lie group representations is that I think there's something interesting going on when you go up one dimension and an instance of it might be in this idea of identifying a knot from a picture taken with a phone. If instead of taking a picture you take a moving video recording as you move the point of view around the knot then you develop a kind of 2-manifold, don't you? I think of it like this because the knot itself is a 1-manifold, and as you turn it around in the embedding space its projection evolves continuously.This doesn't make sense mathematically because the point of view of the camera doesn't change anything about the knot itself which just sits there, but to me there seems to be some sense you could think of the string tracing out a continuous surface in space as it rotates, and if it rotates through 360° and the position is restored at the end then this surface must be closed, so there must be some connection between the knot and subgroups of so(3). I do remember you mentioning that ICM 2018 talk Knots, 3-Manifolds and Instantons by Kronheimer and Mrowka but it's a bit over my head!

 

Of course the surface you get depends upon where the camera is pointing as it rotates. If you think of the circle rotating about an axis through its centre you get a sphere, but if the axis of rotation is outside the circle you get a torus. But the surfaces all together still must represent the whole knot embedding, mustn't they? Or are they just the boundaries of the knot complement?

Maybe of you turn the problem around and consider what paths of a single point moving about as you rotate the frame generate you can get some idea of the problem? Think of a knot as the closed curve traced out by a single point moving in a continuous space as a process where a 0-manifold is developed in a 1-manifold, and the 1-manifold develops into  a 2-manifold and that can happen in various ways depending upon how the camera moves around, ... I think that just makes it a harder problem, though, doesn't it? But somehow all these are just ways of looking at the same unchanging knot, ...  

See Keenan Crane's Discrete Differential Geometry Course. It's quite easy to develop discrete simplicial meshes from these primitives, but I don't know how quickly you can compute the topological invariants of the surfaces you develop that way because they might be rather large. See also Knots and surfaces - the fascinating topology of n-manifolds.

People doing automated theorem proving ought to be interested in this:

 

 

 

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See Daniel Tubbenhauer - How good are (quantum) knot invariants? and in particular the ICM 2018 talk Knots, 3-Manifolds and Instantons by Kronheimer and Mrowka:

See also SU(2) and the quaternions by Qiaochu Yuan.

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