Heegaard splittings of Brieskorn spheres

All Brieskorn spheres are small Seifert fibred spaces (small SFS, in brief), i.e. they admit a fibration $S^1 \to \Sigma(p,q,r) \to S^2$ with three multiple fibres. This is easier to see when $p,q,r$ are pairwise coprime: the fibration come from the action of $S^1\subset \mathbb{C}$ on $\Sigma(p,q,r)$ given by $\theta\cdot(x,y,z) = (\theta^{qr}x, \theta^{rp}y, \theta^{pq}z)$

Each small SFS $M$ admit a genus-2 Heegaard splitting; for instance, take two singular fibres of $M$ and an arc in $M$ that lifts a simple arc connecting the images of the two singular fibres. A neighbourhood of the two fibres and the arc is a 2-handlebody, whose complement is also a 2-handlebody, so we have a Heegaard decomposition of $M$. This is called a vertical Heegaard splitting of $M$.

Actually, genus-2 Heegaard splittings have been classified independently by Boileau, Collins, and Zieschang (Ann. Inst. Fourier 41 no. 4, 1991) and by Moriah (Invent. Math. 91, 1988).