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

TL;DR

This paper proves topological constraints on Lagrangian intersections for certain knots in ^3, using symplectic field theory and algebraic properties of augmentation varieties over , revealing limitations on Hamiltonian diffeomorphisms.

Contribution

It introduces a novel algebraic constraint on augmentation varieties over  that restricts Lagrangian intersections for specific knots, combining symplectic topology and arithmetic methods.

Findings

No compactly supported Hamiltonian diffeomorphism can make certain conormal bundles intersect the zero section cleanly along the unknot.

The algebraic constraint on the augmentation variety over  is valid only when  is not algebraically closed.

The proof employs an arithmetic argument over =_q to establish topological restrictions.

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 $\mathbf{k}=\mathbb{Q}$.