Estimating the number of Reeb chords using a linear representation of the characteristic algebra

Abstract

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-Sullivan and Ekholm-Etnyre-Sabloff which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $\Lambda$ of $\mathbb C^{n}\times \mathbb R$, $n \ge 1$, 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 that is possible to achieve with a Legendrian submanifold whose Chekanov-Eliashberg algebra is acyclic.