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Comment by lordnacho | Hacker News Reader

I saw a clip from the Big Bang Theory the other day, where it was mentioned that there are no knots in 4 dimensions and higher. I wondered why this is so and found this elegant "proof":

https://abel.math.harvard.edu/archive/21a_spring_06/exhibits...

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nice proof!

usually, "higher dimensional knot" refers to embeddings of n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if the distinction between R^(n+2) and (n+2)-spheres scares you, don't worry about it! it's just one point!)

usual knot theory: n = 1, m = 3 OP's proof relates to: n = 1, m = 4

when m - n (the "codimension") is >2, as in the the case from OP's post, there is "so much room" that unknotting can always happen. and at codimension 1, there "isn't enough room". so the interesting theory is codim-2.

in fact, there is a well studied theory. here's a book on the subject (disclaimer: I haven't read it): https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf

That's how I like to explain it to people, though there's a small caveat that doesn't really affect the argument, but it's worth considering. The only times a knot-with-hues actually corresponds to a knot in Euclidean 4-dimensional space are when you can smoothly modify just the colors to make the knot be monochromatic. For example, if the knot goes through the whole color wheel of hues, 0 to 360 degrees, then that corresponds to a knot in a different space (R^3 x S^1). (A way to avoid this problem is to not use the color wheel, but instead use, say, wavelength of a spectral color.)

Although there are no nontrivial circle knots (S^1 knots) in R^4, there are nontrivial sphere knots (S^2 knots). That well-advertised Quanta article about Lisa Piccirillo's work is about this sort of thing.

Could this be a reason why space happens to be 3 dimensional? Like, I don't know, maybe knots are somehow important to the structure of the universe and it couldn't exist with some other dimensionality? Whenever these "magic numbers" appear in math it always makes me curious.

I read that three-dimensional space is the only one where gravitational orbits are possible, because inverse-cube / inverse-linear etc. doesn't have orbits stable to minor perturbations.

From you link:

> You would not be able to tie a shoe in four dimensional space.

I'm not convinced. The thing is that if you tie your shoes with bows at the end, those bows are knot-theoretically trivial, even in 3d-space.

Still a nice proof!

so knots only exist when there are exactly 3 dimensions? That's interesting.

The fourth dimension is enough to untangle a knot made out of a one dimensional space (a loop of string). But you can make two dimensional knots in four dimensional space: this is a surface that is knotted with itself in four dimensions. And probably the pattern extends to higher dimensions.

Even simpler, just take a 3D knot, and pull on it. Where it gets stuck, just lift a strand up in the 4th dimension.

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