Contact homology and one parameter families of Legendrian knots

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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Invariants of Legendrian Knots in Circle Bundles

Communications in Contemporary Mathematics ◽

10.1142/s0219199703001075 ◽

2003 ◽

Vol 05 (04) ◽

pp. 569-627 ◽

Cited By ~ 11

Author(s):

Joshua M. Sabloff

Keyword(s):

Contact Structure ◽

Homology Class ◽

Graded Algebra ◽

Legendrian Knots ◽

Contact Homology ◽

Circle Bundle ◽

Differential Graded Algebra ◽

Circle Bundles ◽

Standard Contact ◽

Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

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The contact homology of Legendrian knots with maximal Thurston–Bennequin invariant

Journal of Symplectic Geometry ◽

10.4310/jsg.2013.v11.n2.a2 ◽

2013 ◽

Vol 11 (2) ◽

pp. 167-178 ◽

Cited By ~ 6

Author(s):

Steven Sivek

Keyword(s):

Legendrian Knots ◽

Contact Homology

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Product structures for Legendrian contact homology

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004110000460 ◽

2010 ◽

Vol 150 (2) ◽

pp. 291-311 ◽

Cited By ~ 10

Author(s):

GOKHAN CIVAN ◽

PAUL KOPROWSKI ◽

JOHN ETNYRE ◽

JOSHUA M. SABLOFF ◽

ALDEN WALKER

Keyword(s):

Duality Theorem ◽

Higher Order ◽

Legendrian Knots ◽

Contact Homology ◽

Massey Products ◽

Cup Product ◽

Classical Invariant ◽

Fourth Author ◽

Product Structures ◽

Legendrian Contact Homology

AbstractLegendrian contact homology (LCH) is a powerful non-classical invariant of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and non-commutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products – and, more generally, an A∞ structure – on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

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CYLINDRICAL CONTACT HOMOLOGY OF A DEHN TWIST

International Journal of Mathematics ◽

10.1142/s0129167x09005819 ◽

2009 ◽

Vol 20 (12) ◽

pp. 1479-1525

Author(s):

MEI-LIN YAU

Keyword(s):

Dehn Twist ◽

Open Book ◽

Contact Homology

We use open book representations of contact 3-manifolds to compute the cylindrical contact homology of a Stein-fillable contact 3-manifold represented by the open book whose monodromy is a positive Dehn twist on a torus with boundary.

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