Symplectically knotted codimension-zero embeddings of domains in R4

Abstract

We show that many toric domains $X$ in ${\mathbb{R}}^4$ admit symplectic embeddings $\varphi$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $\varphi \left(X\right)$ 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 $B^4\left(1\right)$; by contrast a result of McDuff shows that $B^4\left(1\right)$ (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 $S^1$-equivariant symplectic homology.