observed above, $HF^{\infty }({Y_{m,n}},\mathfrak{s})$ has two ${\mathbb{F}}[U,U^{-1}]$ summands, so the differential $\partial ^{\infty }$ in this class has to be trivial. This means there is an even number of holomorphic disks from $[\{x_{r},b_{j},f_{2}\},i]$ to $[\{x_{r},b_{j},f_{1}\},i]$. There is a disk from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ with a bigonal domain $D_{13}$. By the Riemann mapping theorem, this disk has a unique holomorphic representative. The domain of any disk $\phi $ from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ is given by $\mathcal{D}(\phi )=D_{13}+a\mathcal{P}+b\Sigma $ for two integers $a$ and $b$. Such a disk has Maslov index $\mu (\phi )=1+2b$ and $n_{z}(\phi )=b$. It follows that the Maslov index equals 1 if and only if $n_{z}(\phi )=b=0$, which implies that the domain $D_{13}+a\mathcal{P}$ has only non-negative multiplicities when $a\in \{0,1\}$. Thus the domain of the second holomorphic disk from $\{x_{r},b_{j},f_{2}\}$ to $\{x_{r},b_{j},f_{1}\}$ is $D_{13}+\mathcal{P}$. This disk has an odd number of holomorphic representatives. The differential of the chain complex $CF^{+}({Y_{m,n}})$ in this class is trivial: $\partial ^{+}[\{x_{r},b_{j},f_{2}\},i]=\partial ^{+}[\{x_{r},b_{j},f_{1}\},i]=0$ and it follows that $$HF^{+}({Y_{m,n}},\mathfrak{s})={\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$$ is freely generated by the elements $[\{x_{r},b_{j},f_{2}\},i]$ and $[\{x_{r},b_{j},f_{1}\},i]$ for $i\geq 0$.

**Class $\mathfrak{s}_{0}+\mu _{1}$** $$\xymatrix{ \{x_{m},c_{1},e_{1}\} \ar@{->}[d] & \quad \\
\{y_{1},c_{1},d_{1}\} & \{x_{m-2},b_{1},f_{2}\} \ar@{.>}[d]\\
\quad & \{x_{m-2},b_{1},f_{1}\}}$$

[1]{} & {x\_[m]{},c\_[1]{},e\_[1]{}}{x\_[m-2]{},b\_[1]{},f\_[2]{}}D\_[1]{}+D\_[2]{}+D\_[3]{}+D\_[6]{}+D\_[7]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}-\
& -D\_[13]{}+D\_[16]{}+A\_[1]{}+…+A\_[m-3]{}+B\_[1]{}+…+B\_[n-3]{},\
& 