Cellular Legendrian contact homology for surfaces, part II

This paper is a continuation of [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)]. For Legendrian surfaces in [Formula: see text]-jet spaces, we prove that the Cellular DGA defined in [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)] is stable tame isomorphic to the Legendrian contact homology DGA, modulo the explicit construction of a specific Legendrian surface. In [Cellular computation of Legendrian contact homology for surfaces, to appear in Internat. J. Math.], we construct this surface, thereby completing Theorem 5.1 and the proof of the isomorphism.

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Morse flow trees and Legendrian contact homology in 1–jet spaces

Geometry & Topology ◽

10.2140/gt.2007.11.1083 ◽

2007 ◽

Vol 11 (2) ◽

pp. 1083-1224 ◽

Cited By ~ 17

Author(s):

Tobias Ekholm

Keyword(s):

Jet Spaces ◽

Contact Homology ◽

Legendrian Contact Homology

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Cellular Legendrian contact homology for surfaces, part III

International Journal of Mathematics ◽

10.1142/s0129167x1950037x ◽

2019 ◽

Vol 30 (07) ◽

pp. 1950037

Author(s):

Daniel Rutherford ◽

Michael Sullivan

Keyword(s):

Gradient Flow ◽

Contact Homology ◽

Cellular Computation ◽

Legendrian Submanifolds ◽

Legendrian Contact Homology ◽

Differential Geom

This paper is a continuation of [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.]. We construct by-hand Legendrian surfaces for which specific properties of their gradient flow trees hold. These properties enable us to complete the proof in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.] that the Cellular DGA defined in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part I, preprint (2016), arXiv:1608.02984] is stable tame isomorphic to the Legendrian contact homology DGA defined in [T. Ekholm, J. Etnyre and M. Sullivan, The contact homology of Legendrian submanifolds in [Formula: see text], J. Differential Geom. 71(2) (2005) 177–305].

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Legendrian contact homology and topological entropy

Journal of Topology and Analysis ◽

10.1142/s1793525319500031 ◽

2019 ◽

Vol 11 (01) ◽

pp. 53-108 ◽

Cited By ~ 1

Author(s):

Marcelo R. R. Alves

Keyword(s):

Growth Rate ◽

Topological Entropy ◽

Infinite Family ◽

Contact Structures ◽

Legendrian Knots ◽

Contact Homology ◽

Contact Manifolds ◽

3 Dimensional ◽

Reeb Flows ◽

Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

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