Let $Y^3$ be a 3-manifold obtained by surgery on $S^3$ along hopf-link with framing $p,q\in \mathbb{Z}$.
I know that $Y^3\cong L(pq-1,p)$ from the Rolfsen twist.
But, I wonder how can I compute the fundamental group from the surgery diagram directly. (e.g. using group presentation?)
Thanks.