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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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What's about "quantum modular forms"?
There is an article Conformal Field Theory and Torsion Elements of the Bloch Group by W. Nahm in the book Frontiers in Number Theory, Physics, and Geometry II. This is a reference for one direction in …