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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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How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In 2011 Coward-Lackenby proved the following, and they also provided an upper bound on the number of moves. There is a computable function F : N×N → N such that for any two connected diagrams D …
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