Torus knots in Lens spaces, open Gromov-Witten invariants, and topological recursion

Starting from a torus knot $\mathcal{K}$ in the lens space $L(p,-1)$, we construct a Lagrangian sub-manifold $L_{\mathcal{K}}$ in $\mathcal{X}=\big(\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)\big)/\mathbb{Z}_p$ under the conifold transition. We prove a mirror theorem which relates the all genus open-closed Gromov-Witten invariants of $(\mathcal{X},L_{\mathcal{K}})$ to the topological recursion on the B-model spectral curve. This verifies a conjecture in \cite{Bor-Bri} in the case of lens space.