i\leq n-2$}\\
& \epsilon (\{b_{n},-\},\{b_{1},-\})=(n-1)\mu _{2}+\mu _{1}=-\mu _{1}-\mu _{2}\\
& \epsilon (\{a_{2},-\},\{a_{1},-\})=\mu _{1}+\mu _{2}-\mu _{3}\\
& \epsilon (\{y_{2},-\},\{y_{1},-\})=\mu _{1}+\mu _{2}+\mu _{3}\\
& \epsilon (\{c_{2},-\},\{c_{1},-\})=\epsilon (\{e_{2},-\},\{e_{1},-\})=\mu _{1}\\
& \epsilon (\{d_{1},-\},\{d_{2},-\})=\epsilon (\{u_{2},-\},\{u_{1},-\})=\mu _{2}\end{aligned}$$

Suppose that $m,n\geq 4$. The equivalence classes of generators in the torsion $\sp $ structures, given by the $\epsilon $-relations above, are given below. For now, we will choose the origin for the $\sp $ structures arbitrarily and call it $\mathfrak{s}_{0}$. We will show later that $\mathfrak{s}_{0}$ is in fact the $\sp $ structure $\mathfrak{s}_{1,n-1}=\mathfrak{s}_{m-1,1}$ (see Lemma \[lemmasp\]).

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for $(r,j)\in \{1,\ldots ,m\}\times \{1,\ldots ,n-1\}\backslash \\ \{(m,n-1),(m-2,1),(1,n-2),(m-1,1),(1,n-1),(m-1,2),(2,n-1),(m,1)\}$.

We require $m,n\geq 4$ so that the $\sp $ equivalence classes listed above are all distinct. This is still true if $m\geq 4$ and $n=3$ or vice versa. By this requirement we also make sure that the intersection form of the double plumbing ${N_{m,n}}$ is positive definite, which will be important in Section \[App\].

Now we consider the differentials 