Hamiltonian knottedness and lifting paths from the shape invariant

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