scholarly journals Invariants of Legendrian Knots and Coherent Orientations

2001 ◽  
Vol 1 (2) ◽  
pp. 321-367 ◽  
Author(s):  
John B. Etnyre ◽  
Lenhard L. Ng ◽  
Joshua M. Sabloff
Keyword(s):  
Legendrian Knots

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
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.

scholarly journals Legendrian knots and exact Lagrangian cobordisms

2016 ◽  
Vol 18 (11) ◽  
pp. 2627-2689 ◽  
Author(s):  
Tobias Ekholm ◽  
Ko Honda ◽  
Tamás Kálmán
Keyword(s):  
Legendrian Knots ◽  
Lagrangian Cobordisms

scholarly journals Contact homology and one parameter families of Legendrian knots

2005 ◽  
Vol 9 (4) ◽  
pp. 2013-2078 ◽  
Author(s):  
Tamas Kalman
Keyword(s):  
Legendrian Knots ◽  
Contact Homology

scholarly journals The nonuniqueness of Chekanov polynomials of Legendrian knots

2005 ◽  
Vol 9 (3) ◽  
pp. 1221-1252 ◽  
Author(s):  
Paul Melvin ◽  
Sumana Shrestha
Keyword(s):  
Legendrian Knots

scholarly journals New Invariants of Legendrian Knots

2001 ◽  
pp. 525-534 ◽  
Author(s):  
Yuri Chekanov
Keyword(s):  
Legendrian Knots

scholarly journals Holonomic and Legendrian parametrizations of knots

2000 ◽  
Vol 09 (03) ◽  
pp. 293-309 ◽  
Author(s):  
Joan S. Birman ◽  
Nancy C. Wrinkle
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups ◽  
Algebraic Solution ◽  
Legendrian Knots

Holonomic parametrizations of knots were introduced in 1997 by Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of Garside. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

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