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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)?

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  • $\begingroup$ does the klien bottle count? I don't think its one but I am not sure. if it is I would like a different one. $\endgroup$ – user58512 Feb 25 '13 at 18:21
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    $\begingroup$ The Klein bottle does not count as a 'knotted sphere' - it's not a sphere in the first place! (That is, there's no diffeomorphism $S^2\to K$, because any such would preserve orientation and while $S^2$ is orientable, $K$ isn't.) It's arguably much closer to a torus than a sphere, anyway... $\endgroup$ – Steven Stadnicki Feb 25 '13 at 18:23
  • $\begingroup$ Have you seen this or this? $\endgroup$ – dtldarek Mar 2 '13 at 10:35
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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. enter image description here

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    $\begingroup$ Very nice picture! I can see that this really is a picture of a 2-sphere, but what kind of invariants would be used to show this 2-sphere is actually knotted? $\endgroup$ – Jason DeVito Mar 8 '13 at 16:29
  • $\begingroup$ @JasonDeVito: The fundamental group of the complement should do the trick, which you can compute using Seifert-van Kampen. $\endgroup$ – Cheerful Parsnip Mar 9 '13 at 15:20
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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:

enter image description here

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    $\begingroup$ Oh man, for the longest time I was staring at the transition from $t=3$ to $t=4$ and seeing nothing more than this. If anyone else is as confused as I was, all that's happening is that they take the two loops and just wind them around each other. $\endgroup$ – Rahul Mar 2 '13 at 3:07
  • $\begingroup$ LOL! No, literally, I mean it. $\endgroup$ – Robert Haraway Mar 2 '13 at 4:22
  • $\begingroup$ +1 for the link. Carter's book looks very good. I'm hooked at the preface: "Milnor's book Topology from the Differentiable Viewpoint is a model for clarity, conciseness, and rigor. The current text might be subtitled Topology from Scott Carter's Viewpoint, and critics will, no doubt, say it is a model for none of the above." :-) $\endgroup$ – Jesse Madnick Mar 2 '13 at 9:50
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Ralph Fox's "A Quick Trip Through Knot Theory" [1] has a lot of examples in section 6.

  1. In Topology of 3-Manifolds and Related Topics, M.K. Fort (ed.), 1962.
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