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Hamiltonian knottedness and lifting paths from the shape invariant
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 11 / November 2023
- Published online by Cambridge University Press:
- 18 September 2023, pp. 2416-2457
- Print publication:
- November 2023
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- Article
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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.
