Picture of a 4D knot

A knot is a way to put a circle into 3-space $S^1 \to \mathbb R^3$ and these are often visualized as 2D knot diagrams.

Can anyone show me a diagram of a nontrivial knotted sphere $S^2 \to \mathbb R^4$ (differentiable)?

Solution 1:

Here is a picture of a 4D knot. Most of the picture is embedded in a $3D$ time slice at time $t=0$. Then as you let $t$ increase the two top boundary circles persist, until you reach $t=1$ when they are capped with a disk. Similarly you cap off the two bottom boundary components with disks as $t$ decreases.

You can see that this construction yields a $2$-sphere since it represents two cylinders joined by a tube with caps added on the cylinder's ends.

Solution 2:

The book How Surfaces Intersect in Space by J. Scott Carter covers this subject well, starting on page 226. You can get a free PDF of this book at http://www.maths.ed.ac.uk/~aar/papers/cartert.pdf . The book Surfaces in 4-Space by Carter, Kamata, and Saito covers this in-depth.

The premise of these books is that surfaces in 4-space can be represented as movies of embedded curves in 3-space, knotted or not.

If you want to see such a diagram Right Now®, then here is the first such diagram from Surfaces in 4-Space, the Fox-Milnor knotted sphere, which I am posting here FOR EDUCATIONAL PURPOSES ONLY, NOT FOR PROFIT: