Knots in N D

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Knots in N D

Postby PatrickPowers » Tue Apr 16, 2019 11:33 am

Knots are closed curves.

In 3D the simplest knot is the circle aka the 1-sphere. In ND the simplest knot is the (N-2)-sphere. This is more obvious if you think of the (N-2)-sphere as the (N-2)-ring.

To make an overhand/trefoil knot you cut the ring, wrap one of the loose ends around the other end, then reattach the two ends. This can be done in any number of dimensions.

Question: If N is even, are there two distinct trefoil knots? That is, is the trefoil knot orientable?

Question: is the 4D torus a knot?
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Re: Knots in N D

Postby Klitzing » Tue Apr 16, 2019 12:20 pm

A curve always is c: t -> (c1(t), c2(t), …, cN(t))T, i.e. some 1-dimensional path within RN.
The (N-2)-sphere OTOH essentially is locally (N-2)-dimensional, i.e. not a curve for N>3.

--- rk
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Re: Knots in N D

Postby wendy » Tue Apr 16, 2019 12:23 pm

A self-knotting fabric, such as described in the question, has n dimensions, if the space has 2n+1 dimensions. A trifoil knot in 4d, even if it is made out of a 4d spheric-rope, will come undone as readily as a loop in 3d, that is laid out as a figure of eight. You pick it up and it comes undone.

So you can have a self-knotting hedrix (2d fabric), in five dimensions, and a self-knotting chorix (3-fabric) in seven dimensions.

Knots in other dimensions are effected by taking prismatic sections of the lesser dimension, supposing enough prism exists for friction preventing it to unravel.
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Re: Knots in N D

Postby PatrickPowers » Tue Apr 16, 2019 9:50 pm

According to Wikipedia:

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere (S2) embedded in 4-dimensional Euclidean space (R4). Such an embedding is knotted if there is no homeomorphism of R4 onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.


Here is a paper on how to construct such knots. http://faculty.tcu.edu/gfriedman/papers/spinHKT.pdf
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