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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.

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Step 1: Compute the fundamental group of the link complement. There's a bunch of ways to do this. You'll want a procedure where the output presentation makes it easy to identify the meridians and longitudes of the link. The Wirtinger presentation is quite good for this.

Step 2: Dehn filling is attachment of $S^1 \times D^2$'s to the boundary. You can think of that as the attachment of a $D^2$ followed by a $D^3$. Attaching $D^3$ does not change $\pi_1$ so you're left with only the $D^2$ attachment. By SvK, this amounts to adding a relator to your presentation from step 1.

I'm pretty sure this is in Rolfsen's textbook. Are you reading that?

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