{x\_[m]{},c\_[1]{},e\_[1]{}}{y\_[1]{},c\_[1]{},d\_[1]{}}D\_[7]{} ,\
& {y\_[1]{},c\_[1]{},d\_[1]{}} {x\_[m-2]{},b\_[1]{},f\_[1]{}}D\_[1]{}+D\_[2]{}+D\_[3]{}+D\_[6]{}+D\_[9]{}-D\_[10]{}+D\_[12]{}+D\_[16]{}+\
& +A\_[1]{}+…+A\_[m-3]{}+B\_[1]{}+…+B\_[n-3]{} ,\
& {y\_[1]{},c\_[1]{},d\_[1]{}}{x\_[m]{},c\_[1]{},e\_[1]{}}D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[8]{}+D\_[10]{}+2D\_[13]{}+D\_[15]{}+A\_[1]{}+\
& +…+A\_[m-3]{}, ++2\
& {-,-,f\_[2]{}}{-,-,f\_[1]{}}D\_[13]{}D\_[13]{}+\
& {x\_[m-2]{},b\_[1]{},f\_[2]{}}{x\_[m]{},c\_[1]{},e\_[1]{}}=D\_[4]{}+D\_[5]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+2D\_[13]{}+D\_[14]{}+\
& +D\_[15]{}, ++2\
& {-,-,f\_[1]{}}{-,-,f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+D\_[15]{}+\
& +A\_[1]{}+…+A\_[m-3]{}+\
& {x\_[m-2]{},b\_[1]{},f\_[1]{}}{y\_[1]{},c\_[1]{},d\_[1]{}}=D\_[4]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+2D\_[10]{}+D\_[11]{}+D\_[13]{}+\
& +D\_[14]{}+D\_[15]{}+

There is a disk from $\{x_{m},c_{1},e_{1},\}$ to $\{y_{1},c_{1},d_{1}\}$ with a rectangular domain and a unique holomorphic representative. There is no disk from $\{x_{m},c_{1},e_{1}\}$ to $\{x_{m-2},b_{1},f_{2}\}$ whose domain would be non-negative, so these disks have no holomorphic representatives. Thus $\partial ^{+}[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$ and $\partial ^{+}[\{y_{1},c_{1},d_{1}\},i]=0$. Similarly we have $\partial ^{\infty }[\{x_{m},c_{1},e_{1}\},i]=[\{y_{1},c_{1},d_{1}\},i]$ and $\partial ^{\infty }[\{y_{1},c_{1},d_{1}\},i]=0$, so $HF^{\infty }({Y_{m,n}},\mathfrak{s}_{0}+\mu _{1})$ is generated by $[\{x_{m-2},b_{1},f_{2}\},i]$ and $[\{x_{m-2},b_{1},f_{1}\},i]$. We already know there is an even number of holomorphic disks from $\{x_{m-2},b_{1},f_{2}\}$ to $\{x_{m-2},b_{1},f_{1}\}$, so $\partial ^{+}[\{x_{m-2},b_{1},f_{2}\},i]=\partial ^{+}[\{x_{m-2},b_{1},f_{1}\},i]=0$. It follows that $$HF^{+}(Y,\mathfrak{s}_{0}+\mu _{1})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$$ is freely generated by the elements $[\{x_{m-2},b_{1},f_{2}\},i]$ and $[\{x_{m-2},b_{1},f_{1}\},i]$ for $i\geq 0$.

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

[1]{} & {x\_[m-1]{},c\_[2]{},e\_[2]{}}{x\_[1]{},b\_[n-2]{},f\_[2]{}}-D\_[2]{}-D\_[6]{}-D\_[7]{}+D\_[10]{}+D\_[11]{}+D\_[12]{}-D\_[13]{}+\
& +D\_[14]{}+D\_[15]{}+D\_[16]{}+B\_[1]{}+…+B\_[n-3]{} ,\
& {x\_[m-1]{},c\_[2]{},e\_[2]{}}{y\_[1]{},c\_[2]{},d\_[2]{}}D\_[10]{} ,\
& {y\_[1]{},c\_[2]{},d\_[2]{}}{x\_[1]{},b\_[n-2]{},f\_[1]{}}-D\_[2]{}-D\_[6]{}-D\_[7]{}+D\_[11]{}+D\_[12]{}+D\_[14]{}+D\_[15]{}+\
& +D\_[16]{}+B\_[1]{}+…+B\_[n-3]{} ,\
& {y\_[1]{},c\_[2]{},d\_[2]{}}{x\_[m-1]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+2D\_[13]{}+D\_[15]{}+\
& +A\_[1]{}+…+A\_[m-3]{}, + +2\
& {x\_[1]{},b\_[n-2]{},f\_[2]{}}{x\_[m-1]{},c\_[2]{},e\_[2]{}}=D\_[1]{}+2D\_[2]{}+D\_[3]{}+D\_[4]{}+D\_[5]{}+2D\_[6]{}+2D\_[7]{}+\
& +D\_[8]{}+D\_[9]{}+2D\_[13]{}+A\_[1]{}+…+A\_[m-3]{}, + +2\
& {-,-,f\_[2]{}}{-,-,f\_[1]{}}D\_[13]{}D\_[13]{}+\
& {x\_[1]{},b\_[n-2]{},f\_[1]{}}{y\_[1]{},c\_[2]{},d\_[2]{}}=D\_[1]{}+2D\_[2]{}+D\_[3]{}+D\_[4]{}+D\_[5]{}+2D\_[6]{}+2D\_[7]{}+D\_[8]{}+\
& +D\_[9]{}+D\_[10]{}+D\_[13]{}+A\_[1]{}+…+A\_[m-3]{}+\
& {x\_[1]{},b\_[n-2]{},f\_[1]{}}{x\_[1]{},b\_[n-2]{},f\_[2]{}}=D\_[1]{}+D\_[2]{}+D\_[5]{}+D\_[7]{}+D\_[8]{}+D\_[10]{}+D\_[13]{}+\
& +D\_[15]{}+A\_[1]{}+…+A\_[m-3]{}+

Using analogous reasoning as above gives the homology $HF^{+}(Y,\mathfrak{s}_{0}+\mu _{2})\cong {\mathcal{T}^{+}}\oplus {\mathcal{T}^{+}}$, freely generated by the elements $[\{x_{1},b_{n-2},f_{2}\},i]$ and 