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

TL;DR

This paper establishes a new lower bound on the number of Reeb chords for certain Legendrian submanifolds using finite-dimensional matrix representations of their characteristic algebra, extending previous results and exploring algebraic limitations.

Contribution

It introduces a novel Arnold-type lower bound for Reeb chords based on characteristic algebra representations, generalizing prior augmentation-based results and analyzing algebraic properties.

Findings

Proves a lower bound for Reeb chords using finite-dimensional matrix representations.

Provides examples where characteristic algebra admits representations but Chekanov-Eliashberg algebra does not.

Constructs Legendrian submanifolds with characteristic algebra failing the rank property.

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.