# fundamental group of $\Bbb R^3 \setminus$ (two linked circles)

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

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 $S^1 \times S^1$...
• @AbhishekShrivastava Regarding visualisation... did you look at the diagram in the Hatcher's book? It may also be helpful to start with some simpler examples: (i) $\mathbb R^3$ minus the origin can be deformation-retracted onto a sphere, (ii) $\mathbb R^3$ minus a circle can be deformation-retracted onto a sphere with the north and south poles joined by a straight line segment, which in turn can be deformation retracted onto a sphere wedge a circle (see Hatcher, same example). In each case, you need to imagine that the space is made of clay, and you're trying to "flatten it out". – Kenny Wong Oct 15 '17 at 13:50