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Calculate the fundamental group of the complement in $\mathbb{R}^3$ of

$\{ (x,y,z) \ | \ y = 0 , \ x^{2} + z^{2} = 1\} \cup \{ (x,y,z) \ | \ z = 0 , \ (x-1)^{2} + y^{2} = 1\}.$

Note: this space is $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$.

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    As @JimConant says, you can add a point to make this $S^3-K$, where $K$ is your two linked circles. This does not change the homotopy type. Choosing another point of $S^3$ to be infinity won't change it either, so you can remove a point from $K$ to get a new (homeomorphic) space, $\mathbb{R}^3-L$, where $L$ is the union of a circle in the xy-plane, and the z-axis. Now the deformation retraction to the torus is obvious.2012-05-16

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