Hamiltonian knottedness and lifting paths from the shape invariant

• Published in: Compositio mathematica Vol. 159; no. 11; pp. 2416 - 2457
• Main Authors: Hind, Richard, Zhang, Jun
• Format: Journal Article
• Language: English
• Published: London, UK London Mathematical Society 01.11.2023; Cambridge University Press
• Subjects: Invariants; Obstructions; Toruses; Hamiltonian knottedness; shape invariant; Lagrangian tori; 53D35; symplectic embedding; 53D12
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1112/S0010437X23007479

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.