3 ms·

> That’s so unintuitive… It's pretty simple, actually. Imagine you have a knot you want to untie. Lay it out in a knot diagram, so that there are just finite

> That’s so unintuitive… It's pretty simple, actually. Imagine you have a knot you want to untie. Lay it out in a knot diagram, so that there are just finitely many crossings. If you could pass the string through itself at any crossing, flipping which strand is over and which is under, it would be easy, wouldn't it? It's only knotted because those over/unders are in an unfavorable configuration. Well, with a 4th spatial dimension available, you can't pass the string through itself, but you can still invert any crossing by using the extra dimension to move one strand around the other, in a way that wouldn't be possible in just 3 dimensions. > Or am I just seeing patterns where there aren’t any? Pretty sure it's the latter.

stouset 1y ago
That makes sense for a 2D rope in 4D space, but I’m not convinced the same approach holds for a 3D ”hyperrope” in 4D space.
- Sniffnoy 1y ago
  I'm not sure what you mean here. This is discussing a 1-dimensional structure embeded in 4-dimensional space. If you're not sure it works for something else, well, that isn't what's under discussion. If you just mean you're just unclear on the first step, of laying the knot out in 2D with crossings marked over/under, that's always possible after just some ordinary 3D adjustments. Although, yeah, if you asked me to prove it, I dunno that I could give one, I'm not a topologist... (and I guess now that I think about it the "finitely many" crossings part is actually wrong if we're allowing wild knots, but that's not really the issue)
- nimih 1y ago
  Your intuition is correct, it doesn't! A "3D hyperrope" is in fact just the surface of a ball[1], and it turns out that you can actually form non-trivial knots of that spherical surface in a 4-dimensional ambient space (and analogously they can be un-knotted if you then move up to 5-dimension ambient space, although the mechanics for doing so might be a little trickier than in the 1d-in-4d case). In fact, if you have a k-dimensional sphere, you can always knot it up in a k+2 dimensional ambient space (and can then always be unknotted if you add enough additional dimensions). [1] note that a [loop of] rope is actually a 1-dimensional object (it only has length, no width), so the next dimension up should be a 2-dimensional object, which is true of the surface of a ball. a topologist would call these things a 1-sphere and a 2-sphere, respectively
  - trollbridge 1y ago
    Any time I am tempted to feel smart, I try to go and study some linear algebra and walk away humbled. I will be spending 20-30 minutes probably trying to understand what you said (and I think you typed it out quite reasonably), but first I have to figure out how... a 3D hyperrope is the same as a surface of a ball...

stouset 1y ago

That makes sense for a 2D rope in 4D space, but I’m not convinced the same approach holds for a 3D ”hyperrope” in 4D space.

Sniffnoy 1y ago
I'm not sure what you mean here. This is discussing a 1-dimensional structure embeded in 4-dimensional space. If you're not sure it works for something else, well, that isn't what's under discussion. If you just mean you're just unclear on the first step, of laying the knot out in 2D with crossings marked over/under, that's always possible after just some ordinary 3D adjustments. Although, yeah, if you asked me to prove it, I dunno that I could give one, I'm not a topologist... (and I guess now that I think about it the "finitely many" crossings part is actually wrong if we're allowing wild knots, but that's not really the issue)
nimih 1y ago
Your intuition is correct, it doesn't! A "3D hyperrope" is in fact just the surface of a ball[1], and it turns out that you can actually form non-trivial knots of that spherical surface in a 4-dimensional ambient space (and analogously they can be un-knotted if you then move up to 5-dimension ambient space, although the mechanics for doing so might be a little trickier than in the 1d-in-4d case). In fact, if you have a k-dimensional sphere, you can always knot it up in a k+2 dimensional ambient space (and can then always be unknotted if you add enough additional dimensions). [1] note that a [loop of] rope is actually a 1-dimensional object (it only has length, no width), so the next dimension up should be a 2-dimensional object, which is true of the surface of a ball. a topologist would call these things a 1-sphere and a 2-sphere, respectively
- trollbridge 1y ago
  Any time I am tempted to feel smart, I try to go and study some linear algebra and walk away humbled. I will be spending 20-30 minutes probably trying to understand what you said (and I think you typed it out quite reasonably), but first I have to figure out how... a 3D hyperrope is the same as a surface of a ball...