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 $S^3$:

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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!



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