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

Solution 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.