Knotted Legendrian surfaces with few Reeb chords

Dimitroglou Rizell, Georgios

doi:10.2140/agt.2011.11.2903

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Knotted Legendrian surfaces with few Reeb chords

Georgios Dimitroglou Rizell

Algebraic & Geometric Topology 11 (2011) 2903–2936

DOI: 10.2140/agt.2011.11.2903

Abstract

For $g > 0$ , we construct $g + 1$ Legendrian embeddings of a surface of genus $g$ into $J^{1} (ℝ^{2}) = ℝ^{5}$ 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 $J^{1} (S^{2})$ from a similar perspective.

Keywords

Legendrian surface, Legendrian contact homology, gradient flow tree, generating function

Mathematical Subject Classification 2010

Primary: 53D42

Secondary: 53D12

References

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

Volume 11, issue 5 (2011)