Structure of higher-genus open-closed Gromov--Witten theory of ${\mathcal{O}}_{{\mathbb{P}}^1}(p-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-p-1)$

We study the closed and open Gromov–Witten potentials of the toric Calabi–Yau threefold $$X_p=\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^1}(p-1)\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-p-1)),p\ge 2.$$ We prove closed and open mirror symmetry under a nonvanishing condition on the torus weights, relating these potentials to topological recursion on the mirror curves. We establish polynomial structures for both the higher-genus closed potentials and the stable open potentials. We also establish double-scaling limits for topological recursion on the mirror curves. In particular, our results for the closed potentials prove the higher-genus ansatz and the double-scaling conjecture of Caporaso–Griguolo–Mariño–Pasquetti–Seminara.