Can A Knot Be Tied In Anything Other Than 3 Dimensions?

Question

I've heard that a knot can only be tied in 3 dimensions.

Does this only apply to 1 dimensional topologies?

What about a 2 dimensional topology in 4 dimensions or a 2, 3, or 4 dimensional topology in 5 dimensions?

Answer 1

A n-dimensional knot can be untied in a n+1-(or more)dimensional space: you have at least one more dimension and the threads can be moved of position through that extra dimension without colliding with the rest of the threads of the knot remaining in the n-dimensional space (since only that part has been moved into the n+1 dimension and the rest has not). Once you move a thread in the n+1-dimension then you can put it back in another position in the original n-dimensional space, untying the knot.

And well of course you can "keep" the knot "tied" in a n+1 dimensional space (and still will be tied for "somebody" looking at it in the same n-dimensional space where the knot exists) if you do not touch it, but it can be easily untied using the extra space (in your question is not clear if you mean that).

That question appears for instance in a very interesting Topology & Geometry lectures (I strongly recommend them!) done by Dr. Tadashi Tokieda, available at YouTube (here)

Answer 2

One analog of knots in higher dimensions is knotted spheres. There are lots of knotted spheres $S^n\hookrightarrow \mathbb R^{n+2}$ for each $n$, in particular knotted $2$-spheres in $\mathbb R^4$ is the next interesting case after knots.

In the smooth case, i.e. assuming your knots have differentiable structures, there are knots in higher codimensions as well.

This reference may be helpful.