Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$

Question

I've been studying Brieskorn manifolds $\Sigma(p,q,r)$ for $p,q,r\geq 2$. I know they are defined as the intersection of the complex surface $z_1^p+z_2^q+z_3^r=0$ as a subset of $\mathbb{C}^3$ and $S^5$, but I also know they are homeomorphic to the $r$-fold cyclic branched cover of $S^3$ branched along the torus link $T(p,q)$. Moreover, I know they are an example of Seifert fibered manifolds $M(b,(a_1,b_1),..., (a_n,b_n))$. I have seen that in Savaliev's "Invariants for homology 3-spheres", he considers the case in which $p,q,r$ are pairwise coprime, in which case $\Sigma(p,q,r)$ becomes a homology 3-sphere. For this case, he gives the condition $a_1\cdots a_n\cdot \sum b_k/a_k = 1+b\cdot a_1\cdots a_n$ to find the Seifert invariants $b_i$ when only the $a_i$'s are known. However, I am interested in the general case when the $a_i$'s may not be pairwise coprime. Is there a similar condition for $p,q,r$ not pairwise coprime that could allow one to find the Seifert invariants for a Brieskorn manifold?

Any help is appreciated, thank you.

Answer 1

This is written in Némethi's book Normal surface singularities, Example 5.1.17. (He actually talks about the more general case of complete intersections of Brieskorn-type.)

He refers to Jankins and Neumann's Lectures on Seifert manifolds and to Neumann and Raymond's Seifert manifolds, plumbing, $\mu$-invariant and orientation-reversing maps.