Knotted symplectic embeddings between domains in R^4

Michael Usher

I will discuss a proof that many toric domains X in R^4 admit symplectic embeddings f into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes f(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 S^1-equivariant symplectic homology. This is joint work with Jean Gutt.

http://scgp.stonybrook.edu/video_portal/video.php?id=3186

Simons- Workshop: Quantitative Symplectic Geometry