2014/09/30 by Georgios Dimitroglou Rizell, Roman Golovko
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Generalization #Matrix (chemical analysis) #Property (philosophy) #Random Matrices and Applications #Rank (graph theory) #Submanifold #Upper and lower bounds #math.SG #msc:53D12 #msc:53D42
paper · pdf · doi:10.2140/agt.2015.15.2887
published as Algebr. Geom. Topol. 15 (2015) 2887-2920 · 28 pages, 6 figures. Final version. (Typos have been corrected.)
openalex publication_date 2015/11/12 · arxiv created 2016/03/09 · arxiv updated 2016/03/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Given a chord-generic, horizontally displaceable Legendrian submanifold 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 . 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 of C n R, n 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 C n R with the property that the characteristic algebra of does not satisfy the rank property. Finally, in the case when a Legendrian submanifold has a non-acyclic Chekanov-Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of . 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.