  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.
