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We develop a non planar topological vertex formalism and we use it to studythe A-model partition function $\mathcal{Z}_{top}$ of topological string on theclass of toric Calabi-Yau threefolds (CY3) in large complex structure limit. Tothat purpose, we first consider the $T^{2}\times R$ special Lagrangianfibration of generic CY3-folds and we give the realization of the class oflarge $\mu $ toric CY3-folds in terms of supersymmetric gauged linear sigmamodel with \emph{non zero} gauge invariant superpotentials $% \mathcal{W}(\Phi) $. Then, we focus on a one complex parameter supersymmetric $U(1) $ gaugedmodel 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 $.

Topological String on Toric CY3s in Large Complex Structure Limit