I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I understand correctly, it is explained that one can realize the homology three spheres by considering a trivial bundle $S^2 \times S^1$ and remove small discs $D_i$ about marked points $x_i ,..., x_n$ on the two sphere. One can then perform a rational Dehn surgery with surgery slopes $a_i/b_i$ along the links given by $\partial D_i \times pt$ to obtain the Brieskorn homology spheres.
More precisely Saveliev shows that there exist $b_i$ that make this Dehn surgery construction into a rational homology sphere and then states that it follows from Kirby calculus that the resulting 3 manifolds are independent of $b_i$. I should say that, unfortunately, I don't know anything about Kirby calculus. Clearly, in this construction the $a_i$ and $b_i$ do not play symmetric roles. Nevertheless, I am interested in the following:
A general question is: To what extent can we determine the homology three spheres from the $b_i$?
A specific question/motivation: I am not at all a low dimensional topologist. However, a Brieskorn homology sphere naturally arises as a surgery of the type above in a construction I am working on. I know that $n=4$ and all $b_i= \pm 1$, or $n=3$ and $b_1, b_2= \pm 1.$ It is natural to ask: what manifold am I possibly looking at?