Symplectically knotted codimension-zero embeddings of domains in R4

We show that many toric domains X in R4 admit symplectic embeddings ϕ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes ϕ(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 B4(1); by contrast a result of McDuff shows that B4(1) (or indeed any 4-dimensional ellipsoid) cannot have this property. The embeddings are constructed based on recent advances in symplectic embeddings of ellipsoids, though in some cases a more elementary construction is possible. The fact that the embeddings are knotted is proved using filtered positive S1-equivariant symplectic homology.