# Open Gromov-Witten Invariants from the Augmentation Polynomial

^{†}

## Abstract

**:**

## 1. Introduction

**Conjecture**

**1.**

#### 1.1. Statement of Main Results

**Proposition**

**1.**

**Proposition**

**2.**

#### 1.2. Organization of the Paper

## 2. Open String Mirror Symmetry and the Augmentation Polynomial

#### 2.1. The Mirror of ${\mathcal{O}}_{{\mathbb{P}}^{1}}(-1,-1)$

**Example**

**1**(Open Gromov-Witten invariants via mirror symmetry).

#### 2.2. Knots and the Conifold Transition

#### 2.3. Open Gromov-Witten Invariants and the Augmentation Polynomial

## 3. Non-Toric Examples

#### 3.1. The ${4}_{1}$ (Figure-Eight) Knot

#### 3.2. The ${5}_{2}$ (Three-Twist) Knot

## 4. Recovering the Augmentation Polynomial

**Lemma**

**1.**

#### 4.1. The Unknot

#### 4.2. The $(3,2)$ Torus Knot

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**The moment polytope and special Lagrangian fibers of X. The moment polytope of $X={\mathcal{O}}_{{\mathbb{P}}^{1}}(-1,-1)$ is its image $\pi \left(X\right)\subset {\mathbb{R}}^{4}$ under the moment map $\pi \left({z}_{i}\right)={\left|{z}_{i}\right|}^{2}$. The images of the special Lagrangian fibers of X in the moment polytope are vertical lines, which can intersect X three ways: along the base ${\mathbb{P}}^{1}$ (type I), along an exterior leg of the polytope (type II), or on a face (type III). Lagrangian fibers of type III have topology ${T}^{2}\times \mathbb{R}$, corresponding to nondegenerate solutions in (4). Fibers of type I and II have topology ${S}^{1}\times {\mathbb{R}}^{2}$, corresponding to degenerate solutions (such as ${c}_{2}=0$ or ${c}_{1}=0$, respectively).

**Figure 2.**The conifold transition and knots. The Lagrangian ${\tilde{L}}_{K}\subset {Y}_{\mu}\cong {T}^{*}{S}^{3}$ is constructed by shifting the conormal bundle of a knot $K\subset {S}^{3}$ off of the zero section. This lift introduces a holomorphic cylinder C connecting the knot on ${S}^{3}$ to its image in ${\tilde{L}}_{K}$. ${Y}_{0}$ is the conifold singularity $xz-yw=0$ in ${\mathbb{C}}^{4}$. The map ${\varphi}_{\mu}:{Y}_{\mu}\to {Y}_{0}$ is a symplectomorphism away from the zero section, so ${\varphi}_{\mu}({\tilde{L}}_{K})$ is a Lagrangian submanifold of ${Y}_{0}$. $X\cong {\mathcal{O}}_{{\mathbb{P}}^{1}}(-1,-1)$ is the small resolution of the conifold singularity, and ${\sigma}_{\u03f5}:X\to {Y}_{0}$ is the corresponding natural map. In fact, there are a family of such maps, where $\u03f5$ parametrizes the symplectic area of the zero section ${\mathbb{P}}^{1}\subset X$. Hence, ${L}_{K}:={\sigma}_{\u03f5}^{-1}\circ {\varphi}_{\mu}({\tilde{L}}_{K})$ is a Lagrangian submanifold of X. The holomorphic disk D is the image of C under the conifold transition.

${\mathit{K}}_{\mathit{d},\mathit{w}}$ | ${\mathit{K}}_{\mathit{d},1}$ | ${\mathit{K}}_{\mathit{d},2}$ | ${\mathit{K}}_{\mathit{d},3}$ | ${\mathit{K}}_{\mathit{d},4}$ |
---|---|---|---|---|

${K}_{0,w}$ | $-1$ | $-\frac{5}{4}$ | $-\frac{28}{9}$ | $-\frac{165}{16}$ |

${K}_{1,3}$ | 2 | 4 | 14 | 60 |

${K}_{2,w}$ | $-2$ | $-\frac{9}{2}$ | $-25$ | $-147$ |

${K}_{3,w}$ | 1 | 0 | $\frac{173}{9}$ | 186 |

${\mathit{K}}_{\mathit{d},\mathit{w}}$ | ${\mathit{K}}_{\mathit{d},1}$ | ${\mathit{K}}_{\mathit{d},2}$ | ${\mathit{K}}_{\mathit{d},3}$ | ${\mathit{K}}_{\mathit{d},4}$ |
---|---|---|---|---|

${K}_{0,w}$ | $-1$ | $-\frac{5}{4}$ | $-\frac{28}{9}$ | $-\frac{165}{16}$ |

${K}_{1,3}$ | 2 | 4 | 14 | 60 |

${K}_{2,w}$ | $-2$ | $-\frac{9}{2}$ | $-25$ | $-147$ |

${K}_{3,w}$ | 1 | 0 | $\frac{173}{9}$ | 186 |

${\mathit{K}}_{\mathit{d},\mathit{w}}$ | ${\mathit{K}}_{\mathit{d},1}$ | ${\mathit{K}}_{\mathit{d},2}$ | ${\mathit{K}}_{\mathit{d},3}$ | ${\mathit{K}}_{\mathit{d},4}$ |
---|---|---|---|---|

${K}_{0,w}$ | $-1$ | $\frac{3}{4}$ | $-\frac{10}{9}$ | $\frac{35}{16}$ |

${K}_{1,3}$ | 0 | $-2$ | 4 | $-12$ |

${K}_{2,w}$ | 2 | 1 | 0 | $\frac{27}{2}$ |

${K}_{3,w}$ | $-1$ | $-4$ | $-12$ | 8 |

${\mathit{N}}_{\mathit{d},\mathit{w}}$ | ${\mathit{N}}_{\mathit{d},1}$ | ${\mathit{N}}_{\mathit{d},2}$ | ${\mathit{N}}_{\mathit{d},3}$ | ${\mathit{N}}_{\mathit{d},4}$ |
---|---|---|---|---|

${N}_{0,w}$ | $-1$ | 1 | $-1$ | 2 |

${N}_{1,3}$ | 0 | $-2$ | 4 | $-12$ |

${N}_{2,w}$ | 2 | 1 | 0 | 14 |

${N}_{3,w}$ | $-1$ | $-4$ | $-12$ | 8 |

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**MDPI and ACS Style**

Mahowald, M.
Open Gromov-Witten Invariants from the Augmentation Polynomial. *Symmetry* **2017**, *9*, 232.
https://doi.org/10.3390/sym9100232

**AMA Style**

Mahowald M.
Open Gromov-Witten Invariants from the Augmentation Polynomial. *Symmetry*. 2017; 9(10):232.
https://doi.org/10.3390/sym9100232

**Chicago/Turabian Style**

Mahowald, Matthew.
2017. "Open Gromov-Witten Invariants from the Augmentation Polynomial" *Symmetry* 9, no. 10: 232.
https://doi.org/10.3390/sym9100232