Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.