  We develop a non planar topological vertex formalism and we use it to study
the A-model partition function $\mathcal{Z}_{top}$ of topological string on the
class of toric Calabi-Yau threefolds (CY3) in large complex structure limit. To
that purpose, we first consider the $T^{2}\times R$ special Lagrangian
fibration of generic CY3-folds and we give the realization of the class of
large $\mu $ toric CY3-folds in terms of supersymmetric gauged linear sigma
model with \emph{non zero} gauge invariant superpotentials $% \mathcal{W}(\Phi
) $. Then, we focus on a one complex parameter supersymmetric $U(1) $ gauged
model involving six chiral superfields ${\Phi_{i}}$ with $\mathcal{W}=\mu
(\prod\nolimits_{i=0}^{5}\Phi_{i}) $ and we use it to compute the function
$\mathcal{Z}_{top}$ for the case of the local elliptic curve in the limit $\mu
\to \infty $.
