TITLE: Calculating the fundamental group of $\mathbb R^3 \setminus A$, for $A$ a circle
QUESTION [10 upvotes]: Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen. 
Any help would be appreciated. Thanks

REPLY [6 votes]: You should definitely check out the Hatcher's Algebraic Topology book page 46. 
It was very hard for me to imagine at first but $\mathbb{R}^3 - S^1$ deformation retracts onto $S^1 \vee S^2$ so just choose $S^1$ and $S^2$ for $C$ and $D$ respectively, since the space is formed as wedge product of two spaces, the intersection is going to be a point only (by definition) whose fundamental group is trivial for sure. Similarly $\pi_1(S^2)$ is also trivial then $\pi_1(\mathbb{R}^3 - S^1)$ is isomorphic to the fundamental group of the circle which is $\mathbb{Z}$.