  We show that many toric domains $X$ in $R^4$ admit symplectic embeddings
$\phi$ into dilates of themselves which are knotted in the strong sense that
there is no symplectomorphism of the target that takes $\phi(X)$ to $X$. For
instance $X$ can be taken equal to a polydisk $P(1,1)$, or to any convex toric
domain that both is contained in $P(1,1)$ and properly contains a ball
$B^4(1)$; by contrast a result of McDuff shows that $B^4(1)$ (or indeed any
four-dimensional ellipsoid) cannot have this property. The embeddings are
constructed based on recent advances on symplectic embeddings of ellipsoids,
though in some cases a more elementary construction is possible. The fact that
the embeddings are knotted is proven using filtered positive $S^1$-equivariant
symplectic homology.
