#### Volume 11, issue 5 (2011)

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Knotted Legendrian surfaces with few Reeb chords

### Georgios Dimitroglou Rizell

Algebraic & Geometric Topology 11 (2011) 2903–2936
##### Abstract

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into ${J}^{1}\left({ℝ}^{2}\right)={ℝ}^{5}$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over ${ℤ}_{2}$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in ${J}^{1}\left({S}^{2}\right)$ from a similar perspective.

##### Keywords
Legendrian surface, Legendrian contact homology, gradient flow tree, generating function
Primary: 53D42
Secondary: 53D12
##### Publication
Received: 4 February 2011
Revised: 23 June 2011
Accepted: 4 August 2011
Published: 23 October 2011
##### Authors
 Georgios Dimitroglou Rizell Department of mathematics Uppsala University Box 480 751 06 Uppsala Sweden