#### Volume 15, issue 5 (2015)

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

### Georgios Dimitroglou Rizell and Roman Golovko

Algebraic & Geometric Topology 15 (2015) 2885–2918
##### Abstract

Given a chord-generic, horizontally displaceable Legendrian submanifold $\Lambda \subset P×ℝ$ 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 ${ℂ}^{n}×ℝ$, $n\ge 1$, 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 $\Lambda \subset {ℂ}^{n}×ℝ$ 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.

##### Keywords
Legendrian contact homology, characteristic algebra, linear representation, Arnold-type inequality
Primary: 53D12
Secondary: 53D42
##### Publication
Received: 29 September 2014
Revised: 6 February 2015
Accepted: 13 February 2015
Published: 10 December 2015
##### Authors
 Georgios Dimitroglou Rizell Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK http://www.dimitroglou.name/ Roman Golovko Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Realtanoda u. 13–15 Budapest H-1053 Hungary http://sites.google.com/site/ragolovko/