Hamiltonian knottedness and lifting paths from the shape invariant

TL;DR

This paper establishes criteria for lifting paths in the Hamiltonian shape invariant of 4D domains, enabling detection of knotted Lagrangian embeddings and providing new obstructions to symplectic embeddings.

Contribution

It offers necessary and sufficient conditions for path lifting in the shape invariant of basic 4D toric domains, advancing understanding of Lagrangian knottedness and symplectic embedding obstructions.

Findings

Criteria for path lifting in shape invariant of toric domains

Detection of knotted Lagrangian tori embeddings

New obstructions to symplectic embeddings

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