Starting from a torus knot K in the lens space L(p,β1), we
construct a Lagrangian sub-manifold LKβ in
X=(OP1β(β1)βOP1β(β1))/Zpβ under the conifold
transition. We prove a mirror theorem which relates the all genus open-closed
Gromov-Witten invariants of (X,LKβ) to the topological
recursion on the B-model spectral curve. This verifies a conjecture in
\cite{Bor-Bri} in the case of lens space.Comment: 43 pages, 6 figure