# zbMATH — the first resource for mathematics

Estimating the number of Reeb chords using a linear representation of the characteristic algebra. (English) Zbl 1330.53107
Summary: 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, Sabloff and Sullivan, which hold for Legendrian submanifolds whose Chekanov-Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $$\Lambda$$ of $$\mathbb{C}^{n}\times \mathbb{R}$$, $$n \geq 1$$, 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 which is possible to achieve with a Legendrian submanifold whose Chekanov-Eliashberg algebra is acyclic.

##### MSC:
 53D12 Lagrangian submanifolds; Maslov index 53D42 Symplectic field theory; contact homology
Full Text:
##### References:
 [1] V I Arnol’d, The first steps of symplectic topology, Uspekhi Mat. Nauk 41 (1986) 3, 229 · Zbl 0618.58021 · mi.mathnet.ru [2] M Audin, Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagragiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988) 593 · Zbl 0666.57024 · doi:10.1007/BF02566781 · eudml:140136 [3] D Bennequin, Entrelacements et équations de Pfaff, Astérisque 107, Soc. Math. France (1983) 87 · Zbl 0573.58022 [4] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 · Zbl 1029.57011 · doi:10.1007/s002220200212 [5] G Dimitroglou Rizell, Legendrian ambient surgery and Legendrian contact homology, preprint (2014) · Zbl 1315.53093 · arxiv:1205.5544 [6] T Ekholm, Y Eliashberg, E Murphy, I Smith, Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal. 23 (2013) 1772 · Zbl 1283.53074 · doi:10.1007/s00039-013-0243-6 [7] T Ekholm, J B Etnyre, J M Sabloff, A duality exact sequence for Legendrian contact homology, Duke Math. J. 150 (2009) 1 · Zbl 1193.53179 · doi:10.1215/00127094-2009-046 [8] T Ekholm, J Etnyre, M Sullivan, Non-isotopic Legendrian submanifolds in $$\mathbbR^{2n+1}$$, J. Differential Geom. 71 (2005) 85 · Zbl 1098.57013 · euclid:jdg/1143644313 [9] T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) 453 · Zbl 1076.53099 · doi:10.1142/S0129167X05002941 [10] T Ekholm, J Etnyre, M Sullivan, Legendrian contact homology in $$P\times\mathbbR$$, Trans. Amer. Math. Soc. 359 (2007) 3301 · Zbl 1119.53051 · doi:10.1090/S0002-9947-07-04337-1 [11] T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms, preprint (2012) · Zbl 1357.57044 · arxiv:1212.1519 [12] T Ekholm, T Kálmán, Isotopies of Legendrian $$1$$-knots and Legendrian $$2$$-tori, J. Symplectic Geom. 6 (2008) 407 · Zbl 1206.57030 · doi:10.4310/JSG.2008.v6.n4.a3 [13] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560 · Zbl 0989.81114 · doi:10.1007/978-3-0346-0425-3_4 [14] Y Eliashberg, E Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013) 1483 · Zbl 1308.53121 · doi:10.1007/s00039-013-0239-2 [15] J Epstein, D Fuchs, On the invariants of Legendrian mirror torus links (editors Y Eliashberg, B Khesin, F Lalonde), Fields Inst. Commun. 35, Amer. Math. Soc. (2003) 103 · Zbl 1044.57007 [16] J B Etnyre, K Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003) 59 · Zbl 1047.57006 · doi:10.1016/S0001-8708(02)00027-0 [17] R Golovko, A note on the front spinning construction, Bull. Lond. Math. Soc. 46 (2014) 258 · Zbl 1287.53069 · doi:10.1112/blms/bdt091 · arxiv:1210.8140 [18] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 · Zbl 0592.53025 · doi:10.1007/BF01388806 · eudml:143289 [19] M B Henry, D Rutherford, Ruling polynomials and augmentations over finite fields, J. Topol. 8 (2015) 1 · Zbl 1312.57033 · doi:10.1112/jtopol/jtu013 [20] K H Ko, S Lee, On Kauffman polynomials of links, J. Korean Math. Soc. 26 (1989) 33 · Zbl 0688.57004 [21] P Lambert-Cole, Legendrian products, preprint (2013) · arxiv:1301.3700 [22] C Leverson, Augmentations and rulings of Legendrian knots, preprint (2014) · Zbl 1361.53065 · arxiv:1403.4982 [23] E Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, preprint (2013) · arxiv:1201.2245 [24] L L Ng, Computable Legendrian invariants, Topology 42 (2003) 55 · Zbl 1032.53070 · doi:10.1016/S0040-9383(02)00010-1 [25] L Ng, D Rutherford, Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra, Algebr. Geom. Topol. 13 (2013) 3047 · Zbl 1280.57019 · doi:10.2140/agt.2013.13.3047 · arxiv:1206.2259 [26] L Rudolph, A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990) 319 · Zbl 0703.57005 · doi:10.1017/S0305004100068584 [27] D Rutherford, Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: The Fuchs conjecture and beyond, Int. Math. Res. Not. 2006 (2006) · Zbl 1106.57012 · doi:10.1155/IMRN/2006/78591 [28] D Sauvaget, Curiosités lagrangiennes en dimension $$4$$, Ann. Inst. Fourier (Grenoble) 54 (2004) 1997 · Zbl 1071.53046 · doi:10.5802/aif.2073 · numdam:AIF_2004__54_6_1997_0 · eudml:116166 [29] S Sivek, The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant, J. Symplectic Geom. 11 (2013) 167 · Zbl 1277.53082 · doi:10.4310/JSG.2013.v11.n2.a2 [30] S L Tabachnikov, An invariant of a submanifold transversal to a distribution, Uspekhi Mat. Nauk 43 (1988) 193 · Zbl 0649.57029 · mi.mathnet.ru
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.