N-Dimensional Knots

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

N-Dimensional Knots

Postby ICN5D » Thu Apr 10, 2014 5:59 am

I remember reading somewhere how knots are impossible in 4D, there is too much room to move in, so it slips past itself, and unties. Furthermore, it read how only odd dimensions can allow knots, even D's make them impossible. This got me thinking about those higher dimensional possible knots, and what they would look like. Or, more so to do with what they're built out of. It seems like it would be possible to tie an entire 2D plane into a linear knot, existing as an elongated region of tied up material. Then, of course expand on this principle: we could then tie a whole 3D cube into a planar knot. This 2D region would be the interlooping area of madness, with crisscrossing 3-planes around in 4 and 5D.
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Re: N-Dimensional Knots

Postby wendy » Thu Apr 10, 2014 7:26 am

trifoil torus does the trick in five dimensions. Start with a trifoil prism, (which is a hedrix in 4d), and connect the ends as a hose.
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Re: N-Dimensional Knots

Postby quickfur » Thu Apr 10, 2014 3:22 pm

Are you sure only odd dimensional space has knots? I don't think that's true. From what I understand, to make a knot in n-dimensional space, the knotting object must be (n-2)-dimensional. This means in 3D (and only 3D!) you can make knots out of (1D) strings. In 4D, you need a 2D sheet in order to make a knot. In 5D, you need a 3D space-sheet in order to make a knot. :lol: And so on. A 1D string in 5D can't be knotted any more than it can be knotted in 4D: the knot will come undone just by pulling on it. You need an actual extended 3D manifold in order to make a real knot in 5D.

In fact, the Klein bottle is an example of a 2D knot in 4D. So is the projective plane IIRC.
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Re: N-Dimensional Knots

Postby ICN5D » Thu Apr 10, 2014 4:12 pm

It was a while ago, my memory could have faded a bit since then :)
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Re: N-Dimensional Knots

Postby anderscolingustafson » Thu Apr 10, 2014 4:26 pm

Do you remember exactly how long ago it was and what the source was?
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Re: N-Dimensional Knots

Postby ICN5D » Thu Apr 10, 2014 6:07 pm

Nope, I'm afraid that, too, has succumbed to memory fade as well! But, it would be funny if I did :)
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Re: N-Dimensional Knots

Postby wendy » Fri Apr 11, 2014 7:55 am

Here is what i know of knots.

1. A self-knotted fabric of X dimensions, lives in 2X+1 dimensions. So you can have a self-knotted latrix in 3D, or a hedrix in 5D, or a chorix in 7D. The trifoil knot is an example of a knotted latrix, although it's not what the seamen are taught. Their knots are structures that maximise friction and generally make it harder to pull two disjoint peices of string. Mathematics supposes that everything is frictionless, so most of the seaman-knots would unpick as readily as the running-threads used to seal up bags.

2. You can use a 'prism-knot' in any dimension, as long as you suppose that friction is not going to hold across the knot. For example, you can have in 3D a reef-knot. If you take this to four dimensions, one would suppose that you are making a 'knot prism', in much the same way that a hedrix (sheet of paper), can be curled into a 'circle prism' or 'spiral prism'. In any case, there are ways of rolling up notes into a spiral prism, and doing something to the prism that ensures safe delivery to the intended reader. Buck-eye couplers on trains are a kind of hook-prism. The pair of hook works in 2d, the third dimension works to make the 2d thicker. One supposes then that friction or something else stops the knot coming undone by lifting one prism relative to the other.

3. Weaving is more than knotting. One does not want the threads to cross each other. In other words, you don't want to be able to shake the cloth and have a bundle of threads at the hem. The only space i have managed this to work is 3d, and in higher dimensions, the prismatic form of it. On the other hand, plants would tend to match this kind of space to optimise the amount of light on the plant.

4. Quickfur mentions the klein bottle in 4D (criss-cap). This is not a knot, but rather a blemish, that works because a hedrix is not a surface in 4D: ie it does not support an 'out-vector'. In 3d, the latrix can be knotted or kinked or whatever, but the latrix (1d space), does not support branching, so the thing has an internal topology equal to a line or circle. With the hedrix, not all hedrices are topologically equal to a sphere or fragment of a flat), one can put any number of 'handles' onto it, without destroying the local 'smoothness' of it. In three dimensions, a hedrix is a surface: it divides inside from outside, and therefore supports an 'out-vector'. In four dimensions, a hedrix does not divide space. So one can have some interesting fancy things like criss-caps (which reverse the outvector, and therefore it can not contain, because the divergence of the contents is the outvector).
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Re: N-Dimensional Knots

Postby quickfur » Fri Apr 11, 2014 2:33 pm

2X+1? That sounds much more optimistic than I've been led to believe. :o Does that apply only to non-frictionless knots? AFAIK mathematical knots (frictionless knots) only work in X+2 dimensions. Or am I wrong in that respect as well?
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Re: N-Dimensional Knots

Postby Keiji » Sat Apr 12, 2014 6:54 am

What Wendy says makes sense (for once ;) ).

It's exactly the same as chaining toratopes. You can make, in 5D, a chain of spheritorispheres ((III)II). That is equivalent to making a knot from a planar surface, say if you were to cut up a number of spheritorispheres and stick them together in arbitrary ways (the spheritorisphere itself would be 5D's equivalent of the unknot). In seven dimensions it works with glomitoriglomes ((IIII)III) and so on.

In four dimensions, you have to chain spheritori with torispheres alternately, or tigers with tigers. Obviously that means you cannot make knots out of spheritori and torispheres since they have to alternate. Although I'm not sure if the possibility of chaining tigers with themselves means you could make some kind of, "1.5D" knot out of... what should we call them, "mutilated tigers" perhaps?

(No actual tigers were harmed in the making of this post :P )
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Re: N-Dimensional Knots

Postby wendy » Sat Apr 12, 2014 9:51 am

Consider a mathematical knot in 3d, and suppose it is the 'x' axis.

In four dimensions, you can project this knot into a knot-prism, where there is now a plain y axis, and the x line is still knotted. You can undo the knot by sufficiently lifting bits against the y-axis, and over another part of the knot.

In five dimensions, you can prevent access to lifting the y-axis, by turning the y-axis into a circle (over the fourth and fifth axis). So you have a knot-circle-prism, which is five dimensional.
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Re: N-Dimensional Knots

Postby anderscolingustafson » Sun Apr 13, 2014 3:36 am

Would tying a sheet into two perpendicular knots that cross through each other to produce a third knot at the intersection between the two knots work in 4d?
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Re: N-Dimensional Knots

Postby Polyhedron Dude » Mon Apr 14, 2014 4:03 am

A "tube" knot should work in 4-D - imagine having a thin tube made of cloth, lets say that the tube is a half inch in diameter, but the thickness is a millimeter, and the tube is a yard long. We could move the circular end towards a place on the tube a few inches in, let the circular part encase the circular section of the inside slice. Buckle around to a part of the tube between the end and the slice and this time squeeze the circular end inside and through this second slice, then go around another part of the tube and then pull - this should make a trifoil knotted region shaped like a ring - it is the lathe of the trifoil knot. This is actually like tying a thin plane (a ribbon) into a knot, but the plane curves into a tube. It may even be possible to crochet in 4-D using tubular yarn. TUBULAR!!! :mrgreen:
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Re: N-Dimensional Knots

Postby quickfur » Mon Apr 14, 2014 4:10 am

Whoa. That's an awesome idea! So tubular 4D yarns would be simultaneously a thread and a ring, so you can knot it like a thread but it holds together like linked rings. :o_o: That's... wow. :o_o:
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Re: N-Dimensional Knots

Postby wendy » Mon Apr 14, 2014 7:45 am

You can make a string of tigers, because there are different kinds of hole in the tiger. One kind of hole allows a whole hedrix to run through it. This is through a single torid-shaped hole. A second kind of hole lets a circle in through one face of a duocylinder and out the other.

So tigers tie their tails, by way of using their holes differently.
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Re: N-Dimensional Knots

Postby Prashantkrishnan » Thu Dec 25, 2014 12:30 pm

In a sense, it is true to say that an x-plane can be knotted in x + 2 dimensions, though another interpretation gives it as 2x + 1 dimensions.

My choice of words is very bad here :( but I don't know how else to explain.
If x = 1, then x + 2 = 2x + 1 = 3, so it does not reveal much.
It just shows that a latrix can be knotted in realmspace.

Wendy has explained how a knotted hedrix in tetraspace would be unstable, and how it can be made stable in pentaspace. We can actually visualise it as a long cylinder in the VWX realm knotted in the VWXYZ pentaspace. The hedrix in question is the curved surface of the cylinder. The process of knotting involves almost forming a duocylinder in the VWXY tetraspace and then, before completing it, moving one of the end points (end circles?! :D ) through the z-direction. Here if we consider the cylinder as the object being knotted, x = 3 and 5 = x + 2 and if we consider the curved surface of the cylinder as the object being knotted, x = 2 and 5 = 2x + 1.

But in this case, it is not possible to knot a hedrix in tetraspace with x = 2 and x + 2 = 4. One might get the wrong notion that if a cylinder could be knotted in pentaspace, a circle could be knotted in tetraspace. But an attempt to knot a circle in tetraspace is as meaningful as an attempt to knot a point in planespace.

Let me now consider the possibilities of knots in hexaspace. Certainly, a prism of the knotted hedrix/cylinder would fail by the same logic as applied in the case of the knotted hedrix in tetraspace. But how about a spherinder being knotted in hexaspace? The surcell of a spherinder is like a prism of the surface of a sphere, giving x = 3, and 6 = x + 3 or 2x. The process for the knot formation is as follows:

A spherinder in the UVWX tetraspace is curved through the UVWXY pentaspace and the formation of a 32 (Number series notation for the Cartesian product of a sphere and a circle) is almost completed when suddenly one of the end spheres is moved through the z-direction and the spherinder is knotted. I have an intuition that this is no more stable than the knotted hedrix in tetraspace but I don't understand why. Can somebody explain?

PS
I hope my terminology is acceptable :)
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Re: N-Dimensional Knots

Postby Teragon » Sun Dec 20, 2015 11:18 am

I think Polyhedron Dude is right, you can form tube knots in 4D, you can well close an unstable knot plane into a stable cylinder mantle in 4D, if the material has a certain elasticity. There's another way of thinking about it: Take a knot of your choice in 3D. Assume the cord with the knot is aligned in the x-direction. Extrude the knot into the w-direction, bend it to the y-direction and close it to a circle in the wy-plane. In the xyz-hyperplane the 4D knot would look like two knots, one the mirror image of each other with the mirror in the xz-plane offset in y by the radius of the cylinder.

There is another possible geometry for 4D knots. We start with the same situation in the xyz-hyperplane, extruding the knot in the w-direction, but close the knot in the wx-plane instead. The result looks like a plane with a "knot-torus" somewhere in the middle. Such a knot could be impossible to unknot if the plane if sufficiently big, because of the growth of the area with the distance from the center of the knot. In the xyz-hyperplane a cut through the knot would look like two knots on the same cord with a mirror plane halfway in between in the yz-plane.

In 5D the corresponding cords would have the shape of a toricylinder mantle (linear extrusion of the surface of a 2-torus) or, even better, a Clifford-torus-cylinder. In the second geometry, the cord would simply be an open 3-string, while the knot would still have the shape of a 2-torus/Clifford torus. And of course there would be a mixture of both, with the cord in the shape of a cubinder mantle.

Are there even more exotic ways to knot hyperstrings or is it in the nature of knots to be essentially three dimensional shapes extended to tori?
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Re: N-Dimensional Knots

Postby Teragon » Sun Dec 20, 2015 11:49 am

anderscolingustafson wrote:Would tying a sheet into two perpendicular knots that cross through each other to produce a third knot at the intersection between the two knots work in 4d?


Wow! I think this actually works. This would be a real 4D knot (2-knot). Could be a mixture of any two 1-knots. In 5D the corresponding knot would look like three perpendicular 2-planes intersecting at a mind-boggling 3-knot.

So in n dimensions 1-knots (1 standing for the number of directions along which the hyperstring makes a knot) are extended in n-3 dimensions. They have to be closed to circles/tori in order to be actual knots (practically they may be stable anyway if the plane is broad enough for the same reason you can't tear apart a paper roll in the direction of its height). 2-knots are intersections of two 1-knots. Generally, in n dimensions m-knots are extended in n-m-2 dimensions.

Ignoring the distinction made in my former post there are three ways to form a stable knot in 5D:
- Extrude a loop-like 1-knot and close it to a 2-torus/Clifford torus.
- Extrude a 2-knot (two 1D 1-knots intersecting) and close it to a loop.
- Extrude a 2-knot lineary and intersect it with a 2D 1-knot to form a 3-knot.

For n dimensions there are n-2 possible such combinations.

Another interesting issue is the possibility of rotating knotted cylinder mantles or 2-spheres against each other if the friction is small enough. As knots build upon friction it might not be as simple as to rotate 4D chain links, a tiger/3-torus and a torisphere, against each other.

To sum it up:
In 2D knots and chains are impossible, hooks (links that can be opened by deforming the materials alone) are possible.
In 3D knots chains and hooks are possible and both parts of a hook can be rotated against each other in certain geometries.
In 4D knots chains and hooks are possible and and all of them can be made in a way that the two linked parts can be rotated against each other. Hooks may have two degrees of freedom for rotations.
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