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Knotted Legendrian surfaces with few Reeb chords. (English) Zbl 1248.53073
In the article under review the author constructs \(g+1\) (for \(g>0\)) Legendrian embeddings of a surface of genus \(g\) into \(J^1(\mathbb R^2)=\mathbb R^5\) which lie in pairwise distinct Legendrian isotopy classes and which all have \(g+1\) transverse Reeb chords (\(g+1\) is the conjectured minimal number of chords). Furthermore, the author shows that for \(g\) of the \(g+1\) embeddings the Legendrian contact homology DGA does not admit any augmentation over \(\mathbb Z/2 \mathbb Z\), and hence cannot be linearized. The author also investigates these surfaces from the point of view of the theory of generating families. Finally, he considers Legendrian spheres and planes in \(J^1(S^2)\) from a similar perspective.

MSC:
53D42 Symplectic field theory; contact homology
53D12 Lagrangian submanifolds; Maslov index
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