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Estimating the number of Reeb chords using a linear representation of the characteristic algebra. (English) Zbl 1330.53107
Summary: Given a chord-generic, horizontally displaceable Legendrian submanifold \(\Lambda\subset P\times \mathbb{R}\) with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on \(\Lambda\). This result is a generalization of the results of Ekholm, Etnyre, Sabloff and Sullivan, which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds \(\Lambda\) of \(\mathbb{C}^{n}\times \mathbb{R}\), \(n \geq 1\), whose characteristic algebras admit finite-dimensional matrix representations but whose Chekanov-Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold \(\Lambda\subset \mathbb{C}^{n}\times \mathbb{R}\) with the property that the characteristic algebra of \(\Lambda\) does not satisfy the rank property. Finally, in the case when a Legendrian submanifold \(\Lambda\) has a non-acyclic Chekanov-Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of \(\Lambda\). These bounds are slightly better than the number of Reeb chords which is possible to achieve with a Legendrian submanifold whose Chekanov-Eliashberg algebra is acyclic.

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