Open-closed duality in higher genus and winding

Let $X$ be a toric Fano surface. Let $π: \widehat{X} \rightarrow X$ be a toric blow up at a point. We use the Topological Vertex [AKMV] to prove a higher genus, higher winding, open-closed duality between the toric Calabi-Yau 3-folds $K_X, K_{\widehat{X}}$. For $g \geq 0, w \geq 1$, we show the equa