How do I prove this using van-Kampen theorem informally ? (2) The second( and the last) problem is this 

Could someone please help me how to calculate $\pi_1(X)$?
 A: An informal calculation might go as follows. First, let's "push" one of the circles to infinity so that we're instead removing a copy of $S^1$ from $\mathbb{R}^3$ and a copy of $\mathbb{R}$ which 'goes through' the circle and goes off towards infinity along the $z$-coordinate. You should hopefully be able to see that this space is a kind of 'maximally fattened up torus' in $3$-space. That is, if we just start expanding the torus as much as we can in all direction in $\mathbb{R}^3$, the only bits which we would not be able to 'fill in' by this fattening process would be a circle inside the complement of the torus, and a line going through the 'hole' in our torus. So our space should have the same fundamental group as the torus, namely $\mathbb{Z}^2$.
To prove this formally, one would need to use Van-Kampen's theorem, to show that $\pi_1(S^3\setminus L)\cong \pi_1(\mathbb{R}^3\setminus L)$ where we view $S^3$ as the one-point compactification of $\mathbb{R}^3$. This makes the 'pushing the circle to infinity' part of the above actually work. The 'fattening up' process is really just saying that this new space deformaiton retracts onto a torus.
A: You also need to understand the intuition of a relation at a crossing, as follows: 

I have demonstrated this to children with a copper tubing pentoil and a nice length of rope. For the connection with the van Kampen theorem, see my book Topology and Groupoids, p. 349. 
