I have to find the fundamental group $\pi_1(X)$ of $X=\mathbb{R^3} \setminus \{$a line $\cup$ a circumference$\}$.
Line passing through the centre of the circle, we have: $\pi_1(X) \cong\left(\mathbb{R^3}\setminus\{0\}\right)\times S^1 \cong \mathbb{Z}\times\mathbb{Z}$
Line "out" from the circumference, using the Van Kampen theorem, $X=A\cup B$ where $A=\mathbb{R^3} \setminus S^1$, $B= \mathbb{R^3} \setminus \{$the line$\}$ and $A\cap B=\mathbb{R^3}$, we have $\pi_1(X) \cong \cfrac{\mathbb{Z}*\mathbb{Z}}{\{e\}} \cong\mathbb{Z}*\mathbb{Z}$
What about the $\pi_1(X)$ when the line tangent to a point on the circle? I have no ideas