## Geometry & Topology

### Rounding corners of polygons and the embedded contact homology of $T^3$

#### Abstract

The embedded contact homology (ECH) of a 3–manifold with a contact form is a variant of Eliashberg–Givental–Hofer’s symplectic field theory, which counts certain embedded $J$–holomorphic curves in the symplectization. We show that the ECH of $T3$ is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves “rounding corners”. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsváth–Szabó Floer homology $HF+(T3)$.

#### Article information

Source
Geom. Topol., Volume 10, Number 1 (2006), 169-266.

Dates
Accepted: 25 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799706

Digital Object Identifier
doi:10.2140/gt.2006.10.169

Mathematical Reviews number (MathSciNet)
MR2207793

Zentralblatt MATH identifier
1101.53053

#### Citation

Hutchings, Michael; Sullivan, Michael G. Rounding corners of polygons and the embedded contact homology of $T^3$. Geom. Topol. 10 (2006), no. 1, 169--266. doi:10.2140/gt.2006.10.169. https://projecteuclid.org/euclid.gt/1513799706

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