Why do we often consider knots to be embedded in $S^3$ instead of $\mathbb{R}^3$?

Question

When doing knot theory toward the end of my algebraic topology course, we often defined knots as embeddings of $S^1$ in $S^3$ instead of $\mathbb{R}^3$. My professor justified this by saying that $S^3$ is just $\mathbb{R}^3$ with a point added at infinity, which I agree with, but it didn't help with any of the proofs we did. He said $S^3$ is compact which makes it nice but I didn't see this 'niceness' show up anywhere. I can see that almost every knot in $S^3$ corresponds to a knot in $\mathbb{R}^3$, except for knots that pass through the point of infinity. Could someone give me an example of where having knots embedded in a compact surface was more useful or convenient than embedding them in $\mathbb{R}^3$?

Answer 1

Knots are commonly studied nowadays using geometry.

In particular, it is important to know whether a knot is hyperbolic, meaning that its complement in $S^3$ has a complete hyperbolic metric of finite volume (this property would be meaningless for the complement of knot in $\mathbb{R}^3$, which never has a complete hyperbolic metric of finite volume).

In the 1970's, Troels Jorgensen and Bob Riley produced examples of hyperbolic knots.

In the late 1970's, William Thurston completely characterized hyperbolic knots in simple topological terms, which was part of a revolution in 3-dimensional geometric topology that he started.