We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r\_1,r\_2,r\_3)$. How should one get from $\Sigma$ to a surgery description of $M$? 



Brieskorn homology spheres
(I'm adding another answer, in case \*cough\* you have no access \*cough\* to the book.)


In practice, you just need to crunch some numbers: you need to find integers $s\_1, s\_2, s\_3$ such that $r\_1r\_2s\_3 + r\_2r\_3s\_1 + r\_3r\_1s\_2 = b$; the surgery presentation is just obtained by doing 0-surgery on the unknot, and $r\_i/s\_i$-surgery on three (pairwise unlinked) meridians. (Maybe you can make a sanity check about the signs by looking at $\Sigma(2,3,5)$.)


This is also described (but not explained) in Casson and Harer's paper *Some homology lens spaces that bound rational homology balls*, which seems to be [freely accessible](https://projecteuclid.org/euclid.pjm/1102734944).


