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

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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 $O\left({n}^{2}\right)$.

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##### Keywords
symplectic embeddings, embedded contact homology
##### Mathematical Subject Classification 2010
Primary: 53D05, 53D42
##### Publication
Revised: 14 November 2017
Accepted: 8 December 2017
Published: 26 April 2018
##### Authors
 Katherine Christianson Department of Mathematics University of California Berkeley, CA United States Jo Nelson Department of Mathematics Columbia University New York, NY United States http://www.math.columbia.edu/~nelson/