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

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