surgery along a null-homologous knot in a 3-manifold, $Z$, and that $W$ is the associated cobordism. The oriented boundary of $W$ is given as $$\partial W= -Z \cup Z_0=-(-Z_0)\cup -Z,$$ indicating that we can view $W$ as a cobordism either from $Z$ to $Z_0$, or from $-Z_0$ to $-Z$. The latter viewpoint will be more relevant for our purposes. Note that $\mathbb{K}(W)={{\mathbb F}}[H^1(Z_0)]$, so that any choice of coefficient module $\mathbb{A}$ chosen for $-Z_0$ will induce the module $\mathbb{A}\otimes_{{{\mathbb F}}[H^1(Z_0)]}\mathbb{K}(W)=\mathbb{A}$ for the Floer homology of $-Z$.

We will primarily specialize to the case where $\mathbb{A}={{\mathbb F}}[C_n]$ is the group algebra over ${{\mathbb F}}$ on the cyclic group $C_n$, though we will also use coefficients in the group algebra on $C_n$ over the Novikov field $\Lambda$. For both, suppose that we are given a Heegaard triple diagram compatible with the cobordism as above, so that it contains a curve representing the $0$-framed longitude. On this curve we place a basepoint $p$. Then for $\psi\in \pi_2({\mbox{\bf x}}, \Theta, {\mbox{\bf y}})$ let $n_p(\partial\psi)$ be the algebraic number of times the boundary of $\psi$ meets the codimension-one submanifold (of the Lagrangian torus) determined by $p$. Taking coefficients in the module ${{\mathbb F}}[C_n]$, where $C_n$ 