Let $ X = \mathbb R^3 \setminus A$, where $A$ represents two linked circles. 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
Best Answer
The complement of two linked circles deformation retracts onto a wedge sum of a sphere and a torus. You should imagine the sphere as a big shell surrounding both circles, and you should imagine the torus as a tube tightly wrapping one circle and threading through the other; the torus lies in the interior of the sphere, and touches the sphere at one point.
This is hard to describe in words, so please see the image in Hatcher, Example 1.23, bottom of page 46.
The problem then reduces to the problem of finding the fundamental group of $S^2 \vee (S^1 \times S^1)$. This can be done using Van Kampen: take the open set $C$ to be an open neighbourhood of the $S^2$ and take $D$ to be an open neighbourhood of the $S^1 \times S^1$...