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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 to motivate the skein relations?

Why they are useful is related to one reason Polynomial invariants are useful themselves: they let you prove theorems, when they acutally do let you prove theorems. In particular, if you have a gadge …
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