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On Newton polytopes of Lagrangian augmentations

Orsola Capovilla-Searle, Roger Casals

Abstract

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings.

On Newton polytopes of Lagrangian augmentations

Abstract

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings.
Paper Structure (17 sections, 15 theorems, 46 equations, 7 figures)

This paper contains 17 sections, 15 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Scientific context
  3. Main results
  4. Preliminaries on the Legendrian dg-algebra
  5. Legendrian links and Lagrangian fillings
  6. Newton polytopes distinguishing Lagrangian fillings
  7. Background on Newton polytopes.
  8. Linear algebra lemmata.
  9. Proof of Theorem \ref{['thm:main']}
  10. Exact Lagrangian cobordisms
  11. Higher-dimensional Legendrians spheres with infinitely many fillings
  12. Two Final Remarks
  13. Top-dimensional simplices as Newton polytopes for augmentations
  14. Distinct non-orientable exact Lagrangian fillings
  15. Non-orientable Lagrangian cobordisms
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Lambda\subset (\mathbb{R}^3, \xi_{std})$ be either $\Lambda(\beta_{11})$ or $\Lambda_1$. Then there exists a $\vartheta$-loop, a Reeb chord $\rho\in\mathcal{A}(\Lambda;\mathbb{Z}_2)$ and an orientable exact Lagrangian filling $L$ with augmentation such that for any $n\in\mathbb{N}$ the Newton polytopes in $\mathbb{R}^{b_1(L)}$ of the augmented values $(\varepsilon_L \circ\mathcal{A}(\varth

Figures (7)

  • Figure 1: The Lagrangian projection of the Legendrian link $\Lambda(\beta_{11})$ associated to the 4-stranded braid $(\sigma_2\sigma_1\sigma_3\sigma_2)^4\sigma_3\sigma_1$. The $\vartheta$-loop for this Legendrian link is obtained by moving the rightmost $\sigma_1$ crossing around the braid, as indicated by the purple dashed arrow.
  • Figure 2: Orientable pinch moves at a Reeb chord $a$, where the crossing $a$ is replaced with its $0$-resolution and two basepoints $s, s^{-1}$. The basepoints are determined by the orientation of the arcs.
  • Figure 3: The Lagrangian projection of a Legendrian knot $\tilde{\Lambda}$ such that there exists a Lagrangian cobordism from $\Lambda(\beta_{11})$ to $\tilde{\Lambda}$ given by pinching the Reeb chord in the shaded region.
  • Figure 4: An example of performing Legendrian surgery on a disk (in purple) with boundary on a spun Legendrian surface, where the vertical arrow indicates the axis of rotation.
  • Figure 5: Non-orientable pinch moves at a Reeb chord $a$, where the crossing $a$ is replaced with its $0$-resolution and two basepoints $s, s^{-1}$. After performing either of the non-orientable pinch moves shown on the left, the arcs may be oriented in either of the four ways shown on the right. The basepoints are determined by the orientation of the arcs.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 32 more