of the three circles $K_{1}$, $K_{2}$ and $K_{3}$. We denote by $\mu _{i}$ and $\lambda _{i}$ the meridian and longitude of the regular neighborhood of $K_{i}$ respectively. Each of the curves $\alpha _{i}$ is homologous to $\lambda _{i}$, and the curve $\beta _{i}$ corresponds to the framing of $K_{i}$ for $i=1,2,3$ (see Figure \[fig:heeg\]).

Thus, the first homology group $H_{1}({Y_{m,n}})$ is given by $$H_{1}({Y_{m,n}})=\left \langle \mu _{1},\mu _{2},\mu _{3}|\, m\mu _{1}+2\mu _{2}=0,\,2\mu _{1}+n\mu _{2}=0\right \rangle ={\mathbb {Z}}\langle \mu _{3}\rangle \oplus T\langle \mu _{1},\mu _{2}\rangle \;.$$ If at least one of the numbers $m,n$ is odd, the torsion group $T$ is cyclic and we get $T\langle \mu _{1},\mu _{2}\rangle ={\mathbb {Z}}_{mn-4}\langle \mu _{i}\rangle $ (if $m$ is odd and $n$ is even then $i=1$, if $n$ is odd and $m$ is even then $i=2$, if both $m$ and $n$ are odd then $i$ could either be $1$ or $2$). If both $m$ and $n$ are even numbers, then $T\langle \mu _{1},\mu _{2}\rangle ={\mathbb {Z}}_{\frac{mn-4}{2}}\langle \mu _{1}\rangle \oplus {\mathbb {Z}}_{2}\langle \frac{m}{2}\mu _{1}+\mu _{2}\rangle $.

[The chain complex]{} \[CF\] We denote the intersections between the $\alpha $ and $\beta $ curves as follows (see Figure \[fig:heeg\]):\
$\alpha _{1}\cap \beta _{1}=\{x_{1},x_{2},\ldots ,x_{m}\}$, $\alpha _{1}\cap 