scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
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.

scholarly journals A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250059 ◽  
Author(s):  
MICHAEL B. HENRY ◽  
DAN RUTHERFORD
Keyword(s):  
Standard Form ◽  
Geometric Interpretation ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Complex Sequence ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
Definition Of ◽  
Morse Complex ◽  
Differential Counts

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

Foliated compressing discs for Legendrian rational tangles

2016 ◽  
Vol 25 (06) ◽  
pp. 1650029
Author(s):  
Gregory R. Schneider
Keyword(s):  
Contact Structure ◽  
Legendrian Knots ◽  
Ongoing Effort ◽  
Rational Tangles ◽  
Standard Contact ◽  
New Framework

We establish a new framework for diagramming both Legendrian rational tangles in the standard contact structure on [Formula: see text] and the signed characteristic foliations of their associated compressing discs, as well as the technical means by which these diagrams can be used to study Legendrian isotopies of such tangles. We then establish a number of results that represent new progress in the ongoing effort to classify Legendrian rational tangles under a pair of operations known as Legendrian flypes. These operations, while topologically isotopies, are known to produce distinct Legendrian objects in many circumstances, a fact that has been of much interest throughout the study and classification of Legendrian knots.

scholarly journals INVARIANTS FOR LEGENDRIAN KNOTS IN LENS SPACES

2011 ◽  
Vol 13 (01) ◽  
pp. 91-121 ◽  
Author(s):  
JOAN E. LICATA
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Riemann Surfaces ◽  
Lens Spaces ◽  
Graded Algebra ◽  
Cyclic Cover ◽  
Legendrian Knots ◽  
Differential Graded Algebra ◽  
Cyclic Group Action

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra [Formula: see text] which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S1-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

THE SUPPORT GENERA OF CERTAIN LEGENDRIAN KNOTS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250105 ◽  
Author(s):  
YOULIN LI ◽  
JIAJUN WANG
Keyword(s):  
Contact Structure ◽  
Open Book ◽  
Legendrian Knots ◽  
Three Sphere ◽  
Open Book Decompositions ◽  
Standard Contact ◽  
Trefoil Knots ◽  
Positive Support

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston–Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not.10 (2010) 1831–1859].

scholarly journals Computing homology invariants of Legendrian knots

2014 ◽  
Vol 23 (11) ◽  
pp. 1450056 ◽  
Author(s):  
Emily E. Casey ◽  
Michael B. Henry
Keyword(s):  
Homology Group ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Combinatorial Interpretation ◽  
Knot Invariants ◽  
Homology Groups ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
The Many ◽  
Morse Complex

The Chekanov–Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov–Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This paper gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.

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 Reeb periodic orbits after a bypass attachment

2013 ◽  
Vol 35 (2) ◽  
pp. 615-672
Author(s):  
ANNE VAUGON
Keyword(s):  
Periodic Orbits ◽  
Contact Structure ◽  
Convex Surface ◽  
Three Dimensional ◽  
Contact Manifold ◽  
Contact Structures ◽  
Contact Homology ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Manifold With Boundary

AbstractOn a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

On contact embeddings of contact manifolds in the odd dimensional Euclidean spaces

2015 ◽  
Vol 26 (07) ◽  
pp. 1550045 ◽  
Author(s):  
Naohiko Kasuya
Keyword(s):  
Contact Structure ◽  
Euclidean Spaces ◽  
Contact Manifolds ◽  
Standard Contact ◽  
Simply Connected

We prove that a closed co-oriented contact (2m + 1)-manifold (M2m + 1, ξ) can be a contact submanifold of the standard contact structure on ℝ4m + 1, if it satisfies one of the following conditions: (1) m is odd (m ≥ 3) and H1(M2m + 1; ℤ) = 0, (2) m is even (m ≥ 4) and M2m + 1 is 2-connected, (3) m = 2 and M5 is simply-connected.

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