contact surgery diagram on Brieskorn manifolds

Question

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a contact structure on the Brieskorn manifold. I want to know how to get a contact surgery diagram of this contact structure. I guess this should have been written down but I could not find any reference.

Answer 1

Here's a partial answer that works when $p=2$. If $\Sigma (a_0,a_1, \dots ,a_n)$ is defined as the link of the the singularity $\sum z_i ^{a_i}$, the map $\pi_0:\Bbb C^{n+1}\to \Bbb C^n$ with $\pi_0(z_0,\dots,z_n)=(z_1,\dots, z_n) $ restricts to an $a_0$-fold branched contact covering $\Sigma(a_0,a_1,\dots,a_n)\to S^{2n-1}$ with branch set $\Sigma(a_1,\dots, a_n)$ which is transverse. This is a theorem of Ozturk and Niederkruger (Thm 6 in the paper here: http://arxiv.org/abs/math/0509660).

In the case of $\Sigma (2,q,r)$, we have $\Sigma (2,q,r)$ is the branched double cover of $S^3$ over $\Sigma (q,r)$ (better known as the $(q,r)$ torus knot). In this paper: http://projecteuclid.org/euclid.jsg/1175790955, Plamenevskaya gives an algorithm to compute the Legendrian surgery diagram of the contact double branched cover over a transverse knot (using a braid diagram of the transverse knot). Of course the knot here is just a torus knot which is already in braid position so you don't have to go through the trouble of finding a transverse representative.

EDIT: It has come to my attention that not to long after the above paper of Plamenevskaya, Harvey-Kawamuro-Plamenevskaya extended this algorithm to give a Legendrian surgery diagram for the $p$-fold contact cover of a transverse link in http://arxiv.org/abs/0712.1557 , so the whole problem is solved. These pictures can get quite complicated.