Open Gromov-Witten Invariants from the Augmentation Polynomial

TL;DR

This paper explores the relationship between open Gromov-Witten invariants and the augmentation polynomial, providing computational methods for specific knots and demonstrating how invariants can be derived from the polynomial.

Contribution

It introduces a method to compute genus zero open Gromov-Witten invariants from the augmentation polynomial for non-toric knots, expanding computational tools in symplectic geometry.

Findings

Computed open Gromov-Witten invariants for figure-eight and three-twist knots.

Derived augmentation polynomial from computed invariants for specific knots.

Established a link between invariants and the augmentation polynomial for certain cases.

Abstract

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.