Link complements in $\mathbb{R}^{3} $ and $S^{3} $

Question

What's the difference between a link complement in $S^{3} $ and a link complement in $R^{3} $? Are they homeomorphic?

Answer 1

They're clearly not homeomorphic as one is compact and the other is not. The main similarity though is that two $\mathbb{R}^3$ knot complements (not link complements) are homeomorphic iff the corresponding knots are isotopic (up to chiral pairings) iff the same knot complements in the one-point compactification (the corresponding link complements in $S^3$) are homeomorphic. So they are as strong as each other as far as being a knot invariant is concerned.

It is also not hard to see that the fundamental groups of both types of link complements will be isomorphic (Van Kampen's theorem). This is called the knot/link group.