4D Knots

[PWrong]

You can't wrap a string around a 4D object.

Split from "Tetronian written language" by iNVERTED

[pat]

Is that possible? You can't wrap a string around a 4D object.

I don't understand why I wouldn't be able to wrap a string around a 4-D object. Take a long, thin, flexible spherinderical string. Wrap it around a stubby, rigid crind. Assuming a little bit of friction, why wouldn't it stay?

[PWrong]

I don't get it either. I think the problem is that you have a string resting on a 3D surface, so it can pretty much move freely, and unwrap. It depends what you mean by wrap. For example, if you wrap a string all the way around a cube, covering four sides like a coil, and then attach the ends at the top and bottom, you can't unwrap it. But in 4D, the string has to fill 6 of the 8 cells on a hypercube. So I guess you'd fill each face with string the way you described. I don't see any reason for it to fall off (except you'd probably need glue rather than friction).

It would just look ugly, and remember that strings can't be tied into knots in 4D.

[pat]

String can be tied into knots in 4-D... it's just that all knots in 4-D are equivalent to the unknot. So... you'd have to either have very "frictionful" string and be willing not to disturb it, or you'd need some other way to attach the ends together.

The more direct answer: If you had a loop of string in 4-D, it would be a loop no matter how it slid or twisted or rotated.

[Keiji]

If you glued a loop of string to a hypercube so that it covered 4 of the cells, you would be able to cut the hypercube in half without touching the string. :wink:

[quickfur]

String can be tied into knots in 4-D... it's just that all knots in 4-D are equivalent to the unknot. So... you'd have to either have very "frictionful" string and be willing not to disturb it, or you'd need some other way to attach the ends together.

The more direct answer: If you had a loop of string in 4-D, it would be a loop no matter how it slid or twisted or rotated.

Interesting. Just today, I saw another mention of this somewhere. Is there any online resource that talks about this in more detail?
Also, I wonder if it's possible to accomplish the same thing by using a 2D sheet instead of a 1D string.... ? :-) Is it possible to "tie" a 2D sheet into a knot in 4D? (Although I'm not sure what that might mean...)

[pat]

Also, I wonder if it's possible to accomplish the same thing by using a 2D sheet instead of a 1D string.... ? :-) Is it possible to "tie" a 2D sheet into a knot in 4D? (Although I'm not sure what that might mean...)

I'm vague on this... but my understanding is that you can knot spheres, but not strips....

[Keiji]

Yes, you can tie knots in planes in 4D.

[pat]

But, they have to be infinite planes, yes/no?

[Keiji]

No, they don't. You can't tie a knot in an infinite plane, because there are no ends. And if you cut the plane, it isn't infinite any more.

[pat]

Well, that doesn't make sense to me. A planar rectangle is, for all purposes, topologically equivalent to a string. You just flatten the string out a little bit, right? And, I thought we all agreed that you can't keep a string knotted.

[quickfur]

What if we knot it in a 2D sense? Something along the lines of twisting the surface in both the +x, -x and +y, -y directions (just like how in 3D we tie a knot by twisting the string along its +x, -x direction). I don't have 4D fingers, so I can't tell you how to do it, but would this make a 4D knot if it could be done?

[Keiji]

Well, that doesn't make sense to me. A planar rectangle is, for all purposes, topologically equivalent to a string. You just flatten the string out a little bit, right? And, I thought we all agreed that you can't keep a string knotted.

If the plane is big enough, it will stick. If it is too narrow, it will slide through. See, tetronians would have to use wide, long but thin "planes" (in double quotes because technically a plane has no thickness) to tie knots.

We have to use long string to tie knots, or otherwise the string will slip through. It's just like that. However, the narrower our string is, the shorter it can be without slipping. Try it with cotton and ordinary 3mm string if you don't believe me.

[pat]

Okay, but that's something different than what I was talking about....

Topologically, a knot (in 3-D) is a loop of string with some amount of crossing of the strings before the loop is closed. A rubber band is a knot. The knot in your shoelace is not a knot unless you fuse together the two loose ends of the lace.

There are knots that one can make in 3-D (after fusing together the free ends of the string) which cannot be undone in 3-D without breaking the string. However, there are no knots that one can make in a string in 4-D (after fusing together the free ends of the string) which cannot be turned into a simple closed loop with no crossings in 4-D. You never have to break the string in 4-D.

So... that's what I'm asking about.... my understanding is that you can make knots in the surface of a 3-D sphere in 4-D (after fusing together any breaks you had to make to tie the knot) such that they cannot be undone in 4-D. Unless I'm mistaken, there's nothing you can do with a rectangle in 4-D that would be particularly different than if you just had a string (unless you're going to fuse together the opposite edges.... then, I suppose you're really tying knots in a torus... which is fine with me. The same argument for a sphere works with any closed (orientable?) surface, afaik.

[Keiji]

Neues Kinder's post from the other topic:

I agree with iNVERTED. Yes, you can tie knots in tetraspace. Except that the simplest knot in tetraspace would be way more complex than the simplest knot in realmspace, because - with there being an extra dimension - it would require more loops for the knot to stay tied.

[wendy]

You can not tie knots out of a line (ie latrix), but you can knot hedrices (2-fabric). In five dimensions, one can also knot hedrices, but the nature of clothing suggests that one ought weave out of chorices (3-fabric).

Knotting by itself suggests simple linkage, while weaving requires a grid of links that do not wander.

[bo198214]

@wendy
Is there a 3d-similar knot arithmetic of hedrice knots in 4 dimensions (or higher)? Especially unique prime factorization?

[wendy]

For actual weaving, one uses a cloth that is two dimensions less than all-space (ie a thread of N-2 space). The reason for this is that the weave must also act as a dividing element, to stop the threads from coming undone. For this, the good old 3d knot theory applies in full force.

For just creating a simple linkage, all you need is to make two surfaces add up to N-1 (eg in 3d, one can make two lines link, or in 5d, two hedrices. In 4d, a latrix + hedrix is required.)

The knot-theory for crossing manifolds like this, is not to me known.

W