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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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Integer surgery on $S^3$

You can think about attaching a solid torus as a two-step process: First, attach $D^2 \times I$ where $D^2$ is the meridional disc of the attaching torus. Secondly, attach the remaining of the soli …
Mehdi Yazdi's user avatar