Hamiltonian knottedness and lifting paths from the shape invariant

Abstract

The Hamiltonian shape invariant of a domain $X \subset \mathbb {R}^4$ , as a subset of $\mathbb {R}^2$ , describes the product Lagrangian tori which may be embedded in $X$ . We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$ -dimensional toric domain such as a ball $B^4(R)$ , an ellipsoid $E(a,b)$ with ${b}/{a} \in \mathbb {N}_{\geq ~2}$ , or a polydisk $P(c,d)$ . As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric $X$ . We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.