# The plumbing graphs of Brieskorn spheres

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?

• Maybe the easy way out is to say that the (normal crossing divisor) resolution of the singularity gives you a plumbing. Jan 29 '20 at 0:04
• Even if we assume that the plumbed $4$-manifold is negative definite, does it makes the problem easier?
– user150450
Jan 29 '20 at 9:47
• 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.) Jan 29 '20 at 10:45

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. 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.