Homology action of a pair-of-pants homeomorphism

Orient the three boundary classes $c_1,c_2,c_3$ so that $c_1+c_2+c_3=0$ in $H_1(N;\mathbb{Z})$. A homeomorphism with orientation sign $ϵ$ and boundary permutation $\sigma$ sends $c_i$ to $ϵc_{\sigma (i)}$. Its trace on the rank-two quotient of ${\mathbb{Z}}^3$ by $c_1+c_2+c_3$ is therefore $ϵ(\#\mathrm{Fix}\sigma -1)$.