Open Gromov-Witten invariants from the augmentation polynomial (Q2333430)

Summary: A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of \(X = \mathcal{O}_{\mathbb{P}^1}(- 1, - 1)\) 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 \(L_K \subset X\) obtained from the conormal bundles of knots \(K \subset S^3\). 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.