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he image of these figures in space seems different only due to the fact that the circle is embedded in $\mathbb{R}^3$ differently (in two ways). In a three-dimensional space one cannot continuously and disjointly deform a nontrivial knot into a flat circle. In a four-dimensional space one can already do so.

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Page-2, Visual Geometry and Topology Anatoly Fomenko

How does adding a dimension help us deform the shape ?

RobPratt
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tryst with freedom
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  • Hint: imagine two flies walking towards one another on the real line. How could they pass by if they were restricted to the real line? What happens if they can move off the real line into the plane? Now do you see how you can undo a knot by exploiting a fourth dimension? – Rob Arthan Jul 21 '22 at 23:20
  • Oh. I see. Good point... tho I don't see how the 4D case goes, I do get how it helps. – tryst with freedom Jul 21 '22 at 23:21
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    Look at one of the crossings in your knot diagram. What we want to do is to bring the segment of the knot at the back up to the front: we can use the 4th dimension to do that just as we can use the 2nd dimension to help the two flies pass by in my first comment. – Rob Arthan Jul 21 '22 at 23:28
  • Technically speaking, what Fomenko wrote is false. There are several ways to correct the claim. – Moishe Kohan Jul 22 '22 at 04:26
  • Could you elaborate? @MoisheKohan – tryst with freedom Jul 22 '22 at 06:45
  • Ah I see @Gerry Myerson. I have self closed my question. I have to go through carefully over the answers, but I will edit this question with questions I have if it comes to be so – tryst with freedom Jul 22 '22 at 06:46
  • https://math.stackexchange.com/questions/3819811/ambient-isotopy-and-isotopy – Moishe Kohan Jul 22 '22 at 14:38

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