# Heegaard Floer homology of a genus two diagram of $S^3$

I am reading the introductory paper "Heegaard diagrams and holomorphic disks" by Ozsváth and Szabó (https://arxiv.org/abs/math/0403029v1, Section 2.2), and I do not understand one of the examples. They compute the homology of the following Heegaard diagram of $$S^3$$:

In order to count holomorphic disks connecting $$x_3 \times y_3$$ to $$x_2 \times y_3$$, they consider the uniformization of $$\Delta$$ as a standard annulus with four marked points on its boundary, corresponding to $$x_2, x_3, y_2, y_3$$. They call $$a$$ the angle of the arc in the boundary connecting $$x_3$$ to $$x_2$$ which comes from $$\alpha_1$$, and $$b$$ the angle of the arc in the boundary connecting $$y_3$$ to $$y_2$$ which comes from $$\alpha_2$$. Then, they consider the one-parameter family of conformal annuli with four marked points obtained from $$\Delta \cup \Gamma$$ by cutting a slit along $$\alpha_2$$ starting from $$y_3$$, where the marked points correspond to $$x_2,x_3$$ and $$y_3$$ counted twice. At this point, they state:

A four-times marked annulus which admits an involution (interchanging the two $$\alpha$$-arcs on the boundary) gives rise to a holomorphic disk connecting $$x_3 \times y_3$$ to $$x_2 \times y_3$$. By analyzing the conformal angles of the $$\alpha$$ arcs in this one-parameter family, one can prove that the mod $$2$$ count of the holomorphic is $$1$$ iff $$a < b$$.

I have many questions. First, I do not really get what $$a$$ and $$b$$ are. I cannot fully understand why the involution gives rise to a holomorphic disk, although I am quite convinced. The thing I do not get at all is the analysis of the conformal angles in the end.

Any help will be greatly appreciated!