Knots In 4D

[Hornbein]

One may often read that mathematical knots (closed curves) can't exist in a 4D Euclidean space. More precisely, they would all "fall apart" and be equivalent to the circle. I say that 2D manifolds in 4D can be knots. Except for the simple 2-sphere I couldn't imagine what such a knot would be like. Here's a video about how to visualize this.



The way this works is: make an ordinary 3D trefoil knot, except open the closed curve in a boring part of the knot. Put the two new endpoints on an axis of rotation. Flatten the knot to 2D, encoding the third dimension with color. (This flattening isn't necessary, I suppose it was done to simplify things for the graphics program.) The idea is that objects with different color may appear to be intersecting but they really aren't because they are separate in the color dimension. Rotate the trefoil around the access of rotation, with each point in the string leaving a colored space-filling trace. The end result is a 2D embedded in 4D version of the trefoil knot.

It seems plausible that all 3D knots can be 4Dized this way. Question : are there 4D knots that can't be created this way? That is, is there a bijection between knots in 3D and those in 4D?

 

[WWGD]

I think they're referring to $S^1$ knots, which can be unknoted in 4D, but, IIRC, surfaces are knotted if there are at least 2 nonisotopic embedding of said surfaces in 4D.

 

[WWGD]

And by Schoenflies' Theorem shows there are no $S^1$ knots in $\mathbb R^2$. Edit: Alexander 's Horned Sphere shows that Schoenflies doesn't extend to $\mathbb R^3$. I think there aren't higher dimensional extensions of it. Maybe @mathwonk can add in.