Hamiltonian knottedness and lifting paths from the shape invariant

• Main Authors: Hind, Richard, Zhang, Jun
• Format: Journal Article
• Language: English
• Published: 10.05.2021
• Subjects: Mathematics - Symplectic Geometry
• DOI: 10.48550/arxiv.2105.04526

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 $\frac{b}{a} \in {\mathbb N}_{\geq 2}$, 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.