TITLE: In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
QUESTION [37 upvotes]: In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that $S^3=\mathbb{R}^3\cup \{\infty\}$ is compact, so working there is "more convenient".

What are some specific benefits or
  conveniences for working in $S^3$
  instead of $\mathbb{R}^3$?

I could think of one example: In knot theory we work in piecewise-linear category, so working in $S^3$ allows for considering finite triangulations of knot complements.
Another example is in studying achirality, where Smith theory for fixed points of periodic maps on the sphere is used. 
Furthermore, some results in these textbooks are stated for knots in either $\mathbb{R}^3$ or $S^3$, but some other results are stated only for knots in $S^3$. 

Are there results that are correct for
  knots in $S^3$ but not for
  knots in $\mathbb{R}^3$?

REPLY [2 votes]: I think it is much more convenient to work in $S^3$ when you want to compute the fundamental group of the complement of a toric knot , than in $R^3.$
In $S^3$ you use the torus containing the knot to  divide $S^3$ into two $S^1 \times D^2$, and then apply Van Kampen theorem.
In $R^3$ the corresponding divison of the complement of the knot is less natural, it is harder to describe and visualize. 
And ofcourse the fundamental group is the same if you take the knot-complement in $R^3$ or in $S^3.$