#### Volume 18, issue 4 (2018)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Symplectic embeddings of four-dimensional polydisks into balls

### Katherine Christianson and Jo Nelson

Algebraic & Geometric Topology 18 (2018) 2151–2178
##### Abstract

We obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P\left(a,1\right)$ into the ball $B\left(c\right)$ for $2\le a\le \left(\sqrt{7}-1\right)∕\left(\sqrt{7}-2\right)\approx 2.549$, extending work done by Hind and Lisi and by Hutchings. Schlenk’s folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one “convex toric domain” to symplectically embed into another introduced by Hutchings (2016). We also observe that the computational complexity of this criterion can be reduced from $O\left({2}^{n}\right)$ to $O\left({n}^{2}\right)$.

##### Keywords
symplectic embeddings, embedded contact homology
##### Mathematical Subject Classification 2010
Primary: 53D05, 53D42