Open Gromov-Witten Invariants from the Augmentation Polynomial

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = OP1(−1,−1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds LK ⊂ X obtained from the conormal bundles of knots K ⊂ S3. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For (r, s) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the (3, 2) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.