The topological vertex is finitely generated

The topological vertex computes all genus amplitudes for the open topological A-model in $\mathbb{C}^3$ with branes on three toric Lagrangians and, via gluing, both open and closed string amplitudes on any toric Calabi-Yau threefold. In this note we show that the topological vertex is generated by boundary interactions of three disks and three annuli. A generalization of this result is shown to hold for the refined topological vertex.