  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.
