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

Abstract

Given a chord-generic, horizontally displaceable Legendrian submanifold $\Lambda \subset P\times ℝ$ 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 ${ℂ}^n\times ℝ$, $n\ge 1$, 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 $\Lambda \subset {ℂ}^n\times ℝ$ 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 it is possible to achieve with a Legendrian submanifold whose Chekanov–Eliashberg algebra is acyclic.