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

TL;DR

This paper demonstrates the existence of symplectic embeddings of certain domains in four-dimensional space that are knotted in a strong sense, using advanced symplectic homology techniques to establish their nontriviality.

Contribution

It introduces new examples of knotted symplectic embeddings of toric domains in R^4, expanding understanding of symplectic embedding complexity.

Findings

Existence of knotted embeddings of polydisks and convex toric domains in R^4.

Construction methods based on recent advances in symplectic embedding theory.

Use of filtered positive S^1-equivariant symplectic homology to prove knottedness.

Abstract

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.