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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.
24
votes
Seifert surfaces of torus knots
This picture--for a (7,2) torus knot--shows a geometric pattern you can extend to any (n,2) torus knot.
The image is part of Figure 18 in the visually rich paper by Jarke van Wijk and Arjeh Cohen: …
3
votes
Accepted
A uniform upper bound for the linking number of periodic orbits of algebraic vector fields
The Lorenz equations are quadratic, and already have an infinite number of distinct knotted and linked orbits. An answer to one of your other questions has good references on Lorenz orbits, but I also …