Symplectically knotted codimension-zero embeddings of domains in $R^4$

Abstract

We show that many toric domains $X$ in $R^4$ 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 $\varphi (X)$ 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.

52 pages, 4 figures

References in corpus (1)

• Symplectic capacities from positive S^1-equivariant symplectic homology

Cited by in corpus (2)

• Symplectic Banach-Mazur distances between subsets of C^n
• Two closed orbits for non-degenerate Reeb flows