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Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.
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Unknotting algorithm in higher dimensions?
The version of the problem in dimensions higher than 4 is undecidable, by work of Nabutovsky and Weinberger:
https://link.springer.com/content/pdf/10.1007/BF02566428.pdf
This is related to the undecid …