3D Knot in 4D

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3D Knot in 4D

Postby Pentoon » Tue Jan 03, 2012 7:28 pm

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.
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Re: 3D Knot in 4D

Postby Keiji » Tue Jan 03, 2012 7:41 pm

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.
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Re: 3D Knot in 4D

Postby quickfur » Tue Jan 03, 2012 8:00 pm

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? :P). In general, to make a knot in N space, the object being knotted must extend in (n-2) dimensions.
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Re: 3D Knot in 4D

Postby Mrrl » Mon Jan 16, 2012 9:52 am

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?
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Re: 3D Knot in 4D

Postby quickfur » Mon Jan 16, 2012 3:33 pm

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?
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Re: 3D Knot in 4D

Postby Mrrl » Mon Jan 16, 2012 8:18 pm

(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
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Re: 3D Knot in 4D

Postby quickfur » Tue Jan 17, 2012 4:10 am

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?
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Re: 3D Knot in 4D

Postby Mrrl » Tue Jan 17, 2012 9:39 am

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)
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Re: 3D Knot in 4D

Postby Keiji » Tue Jan 17, 2012 1:33 pm

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.
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Re: 3D Knot in 4D

Postby Mrrl » Tue Jan 17, 2012 1:52 pm

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;
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Re: 3D Knot in 4D

Postby Keiji » Tue Jan 17, 2012 2:42 pm

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.
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Re: 3D Knot in 4D

Postby Prashantkrishnan » Wed Jan 15, 2014 7:14 pm

Mrrl wrote: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)


Imagine a square sheet of side length 10 units in the XY plane with verices at (0,0,0,0), (10,0,0,0), (10,10,0,0) and (0,10,0,0).
The edge between (10,0,0,0) and (10,10,0,0) is first lifted through the positive z-direction and then through the positive w-direction.
The edge is now in the first hexadecant(?) (I made up the term analogous to quadrant and octant, I don't know whether there already is a term :D )
From the first hexadecant, it enters the fifth hexadecant by crossing the WXY realm and then crosses the XYZ realm to enter the thirteenth hexadecant, by which time a hole would be formed analogous to the first diagram in the following figure:

Image

The edge is then taken to the ninth hexadecant and from there through the XYZ realm in the interior of the hole mentioned above and then again to the first hexadecant. The knot is now fully formed when tightened. It cannot untangle through the x-, z- or w-direction, since it would collide with itself. But it can untangle through the y-direction, since the sheet has a finite width! The only kind of 2D knot in 4D space that would not untangle is one of infinite width, which would be practically impossible! A fifth dimension would only increase its degree of freedom, making it easier to untangle. Therefore, that kind of a knot would be impossible.

2-spheres can link and form a chain in 5D space because there aren't any holes of dimensions higher than 3. One sphere cannot escape through the interior of the other sphere. But in the case of a knot, I have already demonstrated how it would untangle. The 2D sheet escapes through a 3D hole.
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Re: 3D Knot in 4D

Postby wendy » Thu Jan 16, 2014 6:51 am

You can't take a latrix (string of 1d) or hedrix (string of 2d), and make a perfect topological knot. The laws of linking is that dimension of string + 1 = dimension of space. So you can have a single knotted latrix in 3d, and a knotted hedrix in 5d, but nothing in 4d.

On the other hand, you can take real hedrices and do a "prism knot", that is, a 3d knot extended from line to hedrix in 4d. In fact, you can only weave things using "threads" of N-2 space. Weaving supposes that the various threads can not pass each other, which is why it's N-2 rather than 1.
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Re: 3D Knot in 4D

Postby Prashantkrishnan » Thu Jan 16, 2014 1:17 pm

wendy wrote:You can't take a latrix (string of 1d) or hedrix (string of 2d), and make a perfect topological knot. The laws of linking is that dimension of string + 1 = dimension of space. So you can have a single knotted latrix in 3d, and a knotted hedrix in 5d, but nothing in 4d.


How do we have a knotted hedrix in 5D? From what I have written, it cannot be stable due to the extra degree of freedom (see above). Can you explain? And what is a "prism knot"?
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Re: 3D Knot in 4D

Postby Prashantkrishnan » Thu Jan 16, 2014 1:33 pm

In my previous comment, I have explained what I thought was a knotted hedrix in 4D. I proved that that would be unstable, unless it was of infinite length in the y-direction or the whole was bounded in the y-direction. Ther is one way I can explain what I think is the obstacle in the y-direction.

Take the square in the XY plane which I mentioned in the previous comment. Before starting to knot, draw lines parallel to the x- and y-axes. After the knot is completed, the line parallel to the x-axis would be curved, i.e. not flat with respect to the tetraspace it is in. One way I find to cover the hole through which the knot untangles is to curve the y-axis too. But the resultant knotted hedrix and the original flat hedrix would not be isometric. So how is ti possible?
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Re: 3D Knot in 4D

Postby wendy » Fri Jan 17, 2014 7:44 am

A prism knot is simply a cartesian product of a knot (in 3d), and a line segment in 1d. The string used to tie the knot becomes a hedrix.

In real life, you can knot bits of string, because the restrictions on movement, and the roughness of the string combine to hold something fast. In mathematics, one does not count the roughness, so all the knots shown in the picture above simply would come undone.

In four dimensions, one might need an infinite prism-knot, but in five dimensions, a circle-knot prsim might achieve a similar effect. That is, the prism or cartesian product of a circle-surface and a trifoil knot might produce a hedrix that is undo-able. The greater number of degrees of freedom the higher dimensions afford allow you to bend the straight prismin 4d into a circle in five dimensions.

In four dimensions, you just don't have the ordinary hollow-sphere and hollow-circle toruses. There has been discussion on the torotope section on a creature called a 'tiger', which has two torus-shaped holes. One can form a linked chain of tigers.
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