Knot Groups in S^3 vs 3-space
Solution 1:
The space $\mathbb{R}^3\setminus S^1$ doesn't deformation retract onto the torus, but rather a torus with a disk. This is homotopy equivalent to the wedge sum $S^2\vee S^1$, which has the correct fundamental group.
This shows, though, that homotopically there's a difference up in $\pi_2$. That's from the fact that $\mathbb{R}^3\setminus S^1$ is homeomorphic to $S^3$ minus a circle and a point, but $\mathbb{R}^3$ minus a line is homeomorphic to $S^3$ minus a circle. Removing that extra point makes $\pi_2$ be nontrivial.