## 3D Knot in 4D

Ideas involving the use of more than three spatial dimensions. If you want to talk about spacetime go to the Time Dimensions forum instead!

### 3D Knot in 4D

Could somebody help me? I don't understand why a 3D knot cannot exist in 4D. Am I saying that right? I'd like to see a series of pictures that clearly explain it, with a 2D analogy, if possible.
Pentoon
Mononian

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### Re: 3D Knot in 4D

Well, imagine a 3D knot in 3D space, as we are used to.

Now imagine adding a 4th dimension.

You are now free to lift any part of the rope into the fourth dimension, where it can then be moved in any direction of the original 3D space without colliding with the rest of the rope (since only that part has been moved into the fourth dimension and the rest has not). You can then move it back into the third dimension in a different position.

It's the same reason why you can just walk around a river in 4D, why sponge-like objects can seemingly pass through each other in 4D, and so on.

Sadly there is no 2D analogy since knots aren't possible in 2D.

Keiji

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### Re: 3D Knot in 4D

First, let's use less ambiguous terminology so that we don't confuse ourselves over what is being said. The term "3D knot" is ambiguous because there are two things going on here: the dimension of the ambient space (3D), and the dimension of the extensions of the thing being knotted (e.g., a rope is 1D as far as knots are concerned).

So I'm assuming you mean that in 3D (ambient space), a 1D rope can be tied into a knot, but you're wondering why a 1D rope cannot be knotted in 4D?

The easiest way to understand why this is not possible is to imagine that we take a 1D length of rope in 4D and squeeze it between two parallel hyperplanes, so that it essentially only has 3 degrees of freedom. Then it is essentially equivalent to a 1D rope in 3D, so obviously we can tie a knot with it.

Now, the reason the rope is knotted is because at one or more points along its length, it would need to "cross over" another part of itself in order to become untied. Just think of your regular knot in 3D: if a knotted piece of rope can pass through itself at certain critical points, then the knot can be undone without actually untying the knot. Let's call these points "crossing points". The knot is a knot only because the rope can't pass through itself at these crossing points: another part of it is blocking itself, and that's what holds the knot together.

Now suppose that after tying the knot, we remove the confining parallel hyperplanes, so that the rope again has 4 degrees of freedom. Take one of the knot's crossing points. Notice that the part of the rope that's blocking itself lies only in the 3D hyperplane that it was originally confined to. But now that the confinement is removed, the blocking part of the rope can simply be pulled in the 4th direction so that it no longer blocks the movement of the blocked part of the rope. Thus, we can simply pull the blocked part of the rope over to the other side, and thus the knot becomes undone. Intuitively speaking, this means that any knotted 1D rope in 4D can be unknotted simply by pulling its ends: the knot will just untie itself -- it is actually no knot at all.

You may say, well, what if we knot the rope such that the blocked part is itself blocked by another part so that we can't simply move it out of the way in the 4th direction? It turns out that no matter how you try to tie the knot, a 1D piece of rope simply does not occupy enough space in 4D to be able to block every possible movement of other parts of itself. Mathematically speaking, a knotted 1D rope in 4D is topologically equivalent to the "unknot" (i.e., no knot at all). No matter what you do, there's always some direction in which we can pull the rope so that the crossing point becomes undone. The reason for this is that a 1D piece of rope is not enough to block that extra degree of freedom in 4D; there's always a leftover degree of freedom that lets you undo the knot trivially.

The only way a knot can hold together in 4D is if the "rope" is extended not only in 1D, but in 2D. That is, in 4D, the only way you can knot something is to use 2D sheets. One example of a 4D knot is the Klein bottle -- which is actually a misnomer, because it doesn't hold any water in 4D. It's essentially a "knotted sphere", a sphere knotted with itself in such a way that there's no way you can pull it apart back into a "normal" sphere.

And if you think this is very strange, you're right, but wait till you see 5D. In 5D, even 2D sheets won't knot. Instead, you have knotted 3D realms (knotted space, anyone? ). In general, to make a knot in N space, the object being knotted must extend in (n-2) dimensions.
quickfur
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### Re: 3D Knot in 4D

Are you sure about 5D? We can make a chain of 2D spheres there, so why we can't make a knot on 2D sheet?
Mrrl
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### Re: 3D Knot in 4D

Mrrl wrote:Are you sure about 5D? We can make a chain of 2D spheres there, so why we can't make a knot on 2D sheet?

We can make a chain of 2D spheres in 5D? How?
quickfur
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### Re: 3D Knot in 4D

(x-1)^2+y^2+z^2=4, u=v=0 and (x+1)^2+u^2+v^2=4, y=z=0 are connected
Mrrl
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### Re: 3D Knot in 4D

Hmm this is very interesting. Didn't know this before. It seems that the linkage comes from the fact that the sphere must be immersed in at least 3 dimensions, so that gives it a way to always surround at least one point in the other sphere. So in a sense you still need at least (n-2) dimensions in order to knot, but the surface itself may be less than that if it must be immersed in at least (n-2) dimensions.

Or am I off base here?
quickfur
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### Re: 3D Knot in 4D

Wonders are everywhere in this multidimensional world
You need k-dimensional sheet to make a knot in 2*k+1 - dimensional space (if it is possible at all). And you will have troubles in spaces with even number of dimensions, your knots will either unknot by itselves (if dimensions of sheet are less than k) or be not very pleasant-looking (like knot made of wide band in 3d).
And you can use alternating k- and l- dimensional spheres to make a chain in k+l+1-dimensional space (for 4D it will be either 2-spheres and circles or 3-spheres and pairs of points)
Mrrl
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### Re: 3D Knot in 4D

Mrrl wrote:(x-1)^2+y^2+z^2=4, u=v=0 and (x+1)^2+u^2+v^2=4, y=z=0 are connected

I don't get it; what's that got to do with making a chain of spheres?

And what exactly do you mean by a chain of spheres? The only thing I can think of is threading beads on a string, which should be possible in any dimension >= 3.

Keiji

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### Re: 3D Knot in 4D

I mean that two 2D spheres in 5D are connected like 1D rings in 3D chain: you can't move one of spheres away without intersection with another sphere.

Longer chain may consist of spheres

(x-k)^2+y^2+z^2=1/2, u=0, v=0 for all even k;
(x-k)^2+u^2+v^2=1/2, y=0, z=0 for all odd k;
Mrrl
Trionian

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### Re: 3D Knot in 4D

Ooh, I see now. If you start with a sphere bounded in 3D, then add another dimension you can take a circular cross section, repeat again and you get a cross section of two points. Then you place the other sphere in this cross section containing exactly one point, and the result is that the two spheres cannot be pulled apart.

Keiji