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Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations

Yukihiro Okamoto

Abstract

Let $K$ be a knot in $\mathbb{R}^3$ which has the $(2,q)$-torus knot for $q\neq \pm 1$ or the figure-eight knot as a component of connected sum. For its conormal bundle $L_K$ in $T^*\mathbb{R}^3$, we show that there is no compactly supported Hamiltonian diffeomorphism $\varphi$ on $T^*\mathbb{R}^3$ such that $\varphi(L_K)$ intersects the zero section $\mathbb{R}^3$ cleanly along the unknot in $\mathbb{R}^3$. Using symplectic field theory, the proof is reduced to studying the augmentation variety $V_{\mathbf{k}}(K)$ of $K$ over a filed $\mathbf{k}$. The key point of this paper is finding an algebraic constraint on $V_{\mathbf{k}}(K)$ which is valid only when $\mathbf{k}$ is not algebraically closed, and the proof is completed by some arithmetic argument with $\mathbf{k}=\mathbb{Q}$.

Topological constraints on clean Lagrangian intersections from $\mathbb{Q}$-valued augmentations

Abstract

Let be a knot in which has the -torus knot for or the figure-eight knot as a component of connected sum. For its conormal bundle in , we show that there is no compactly supported Hamiltonian diffeomorphism on such that intersects the zero section cleanly along the unknot in . Using symplectic field theory, the proof is reduced to studying the augmentation variety of over a filed . The key point of this paper is finding an algebraic constraint on which is valid only when is not algebraically closed, and the proof is completed by some arithmetic argument with .
Paper Structure (17 sections, 20 theorems, 116 equations, 1 figure)

This paper contains 17 sections, 20 theorems, 116 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Symplectic field theory for exact Lagrangian cobordisms
  3. Preliminaries
  4. Moduli space of pseudo-holomorphic curves with boundary in an exact Lagrangian cobordism
  5. Chekanov-Eliashberg DGA over $\mathbb{Z}[H_2(Y,\Lambda)]$
  6. DGA map associated to an exact Lagrangian cobordism
  7. Application to clean Lagrangian intersections
  8. Chekanov-Eliashberg DGA of the unit conormal bundle of a knot in $\mathbb{R}^3$
  9. Exact Lagrangian cobordism arising from clean intersection
  10. Proof of main results
  11. General case
  12. Proof of Theorem \ref{['thm-1']} for $K'\# T_{(2,2m+1)}$ ($m\neq 0,-1$)
  13. Review of the definition of $HC_0(K)$
  14. On augmentations of $\mathcal{A}^{\mathrm{knot}}_*(K\# T_{(2,2m+1)})$ for $m\geq 0$
  15. Application of $\mathbb{Q}$-valued augmentations
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K$ be $K'\# T_{(2,2m+1)}$ or $K' \# 4_1$, where $K'$ is an arbitrary knot in $\mathbb{R}^3$ and $m\in \mathbb{Z} \setminus \{0,-1\}$. Then, there is no $\varphi\in \mathrm{Ham}_c(T^*\mathbb{R}^3)$ such that $\varphi(L_{K})$ intersects $\mathbb{R}^3$ cleanly along the unknot in $\mathbb{R}^3$.

Figures (1)

  • Figure 1: The shaded region is the image of $u$. $c_x$ and $c_y$ are paths in $L_{K_0}$ and $u(\mathbb{R}\times \{1\})$ is contained in $L_{K_0}$. $u(\mathbb{R}\times \{0\})$ is contained in $\psi(\mathbb{R}^3)$.

Theorems & Definitions (43)

  • Theorem 1.1: Theorem \ref{['thm-main']} and Theorem \ref{['thm-eight']}
  • Theorem 1.2: Corollary \ref{['cor-aug']}
  • Remark 1.3
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • ...and 33 more