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Let $p,q$ and $r$ be positive integers. A Brieskorn sphere is a closed oriented $3$-manifold defined by $$\Sigma(p,q,r) = \{ x^p+y^q+z^r=0 \} \cap S^5.$$

Its fundamental group is well-known due to Milnor. It is always a rational homology sphere. When $p,q$ and $r$ are further chosen pairwise coprime, then it is an integral homology sphere.

In this case, the plumbing graph of a Brieskorn sphere is well-understood, see for example Section 1 of Saveliev's book: Invariants of Homology 3-Spheres.

One needs to find unique integers $b,p',q',r'$ solving the equation \begin{equation} bpqr+p'qr+pq'r+pqr'=-1 \end{equation} where $1\leq p' \leq p-1$, $1\leq q' \leq q-1$ and $1\leq r' \leq r-1$. It is basically done by taking mod of these integers.

How about the rational case? Is it possible to find a unique representation for the plumbing graph associated to Brieskorn spheres?

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  • $\begingroup$ Maybe the easy way out is to say that the (normal crossing divisor) resolution of the singularity gives you a plumbing. $\endgroup$ Jan 29 '20 at 0:04
  • $\begingroup$ Even if we assume that the plumbed $4$-manifold is negative definite, does it makes the problem easier? $\endgroup$
    – user150450
    Jan 29 '20 at 9:47
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    $\begingroup$ The (minimal) negative definite plumbing is unique; this is due to Neumann. (In particular, this tells you that topology determines the singularity, in a way.) $\endgroup$ Jan 29 '20 at 10:45
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You may simply find the surgery diagram of Brieskorn spheres. This is from Özbağcı's lecture notes.

The small Seifert fibered $3$-manifold $M(r_1, r_2, r_3)$ is defined by the following rational surgery diagram.

enter image description here

Then

  1. $- \Sigma(2,3,5) \cong M \left(\frac{-1}{2}, \frac{1}{3},\frac{1}{5} \right) \cong \partial E_8 \cong S^3_1(3_1)$,

  2. $- \Sigma(2,3,4) \cong M \left(\frac{-1}{2}, \frac{1}{3},\frac{1}{4} \right) \cong \partial E_7 \cong S^3_2(3_1)$,

  3. $- \Sigma(2,3,3) \cong M \left(\frac{-1}{2}, \frac{1}{3},\frac{1}{3} \right) \cong \partial E_6 \cong S^3_3(3_1)$.

where $3_1$ denotes the right-handed trefoil, $S^3_n(K)$ denotes the $n$-surgery on the knot $K$ in $S^3$, and $E_k$ denotes the Dynkin diagram.

You may figure out (most of) these homemorphisms by using Kirby calculus.

For the plumbing graphs, you can read the lecture notes of Némethi.

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