I am reading Knots by Burde and Zieschang (2nd edition). In it the authors claim that for a knot $K$ there exists an isomorphism $\pi_1(\mathbb{R}^3 \setminus K)\cong \pi_1(S^3\setminus K)$ that is induced by inclusion.
This should be straightforward however consider the case of the unknot $S^1$. $\mathbb{R}^3\setminus S^1$ deformation retracts to a torus, and if we view $S^1$ in $S^3$ as line in $\mathbb{R}^3$ then we obtain a deformation retraction of $S^3\setminus S^1$ onto $S^1$. Clearly the fundamental groups of the torus and the circle are not the same.
Am I doing something stupid?
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.