# Brieskorn homology spheres

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 $M$?

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 $\Sigma(2,3,5)$.)
Of course, this is not the only way to get surgery descriptions for Brieskorn homology 3-spheres and in fact for each $\Sigma(p,q,r)$, there are infinitely many hyperbolic manifolds which can be filled to give $\Sigma(p,q,r)$ which do not arise as part of the construction mentioned above.