of a single 2-handle addition along a knot, the induced homomorphism is defined by counting holomorphic triangles in a suitable Heegaard triple-diagram. Explicitly, suppose that $Z_i$ are described by pointed Heegaard diagrams $(\Sigma, {\boldsymbol \alpha}, {{\boldsymbol \gamma}^{ i}}, w)$ such that $(\Sigma, {\boldsymbol \alpha}, {{\boldsymbol \gamma}^{ 1}}, {{\boldsymbol \gamma}^{ 2}}, w)$ is an admissible triple diagram describing $W$ and adapted to the knot in the standard way. Then the Floer chain groups for $Z_i$ are generated over the appropriate coefficient modules by intersection points in ${\mathbb{T}_\alpha}\cap{\mathbb{T}_{{{\boldsymbol \gamma}^{ i}}}}$, and $F_W^\mathbb{A}$ is the map induced in homology by the chain map $$F_{W}^\mathbb{A}(U^{-j}\cdot {\mbox{\bf x}})  = \sum_{{\mbox{\bf y}}\in{\mathbb{T}_\alpha}\cap {\mathbb{T}_{{{\boldsymbol \gamma}^{ 2}}}}}\sum_{\psi\in\pi_2({\mbox{\bf x}},\Theta,{\mbox{\bf y}})} \#{{\mathcal M}}(\psi)\, \mathcal{A}(\psi)\,U^{n_w(\psi) - j}\cdot {\mbox{\bf y}},$$ where the sum is over homotopy classes of triangles $\psi$ whose associated moduli space ${{\mathcal M}}(\psi)$ has dimension 0, and $\Theta\in {\mathbb{T}_{{{\boldsymbol \gamma}^{ 1}}}}\cap{\mathbb{T}_{{{\boldsymbol \gamma}^{ 2}}}}$ is a canonical intersection point. Here $\mathcal{A}: \pi_2({\mbox{\bf x}},\Theta,{\mbox{\bf y}})\to \mathbb{K}(W)$ is an “additive assignment” that we now describe in the situations relevant for us; namely, in the case of a $2$-handle cobordism associated to a “zero surgery" (see [@HolDiskFour] for more details, or Section \[trianglesec\] below).

Assume that $Z_0$ is the $3$-manifold resulting from 0-framed 