scholarly journals Legendrian torus knots in S1 × S2

2015 ◽  
Vol 24 (12) ◽  
pp. 1550064 ◽  
Author(s):  
Feifei Chen ◽  
Fan Ding ◽  
Youlin Li
Keyword(s):  
Contact Structure ◽  
Torus Knots ◽  
Tight Contact

We classify Legendrian torus knots in S1 × S2 with its standard tight contact structure up to Legendrian isotopy.

scholarly journals LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

2007 ◽  
Vol 09 (02) ◽  
pp. 135-162 ◽  
Author(s):  
FAN DING ◽  
HANSJÖRG GEIGES
Keyword(s):  
Rotation Number ◽  
Contact Structure ◽  
Analogous Result ◽  
Jet Space ◽  
Legendrian Knots ◽  
Torus Knots ◽  
Oriented Link ◽  
Linking Number ◽  
Tight Contact ◽  
Knots And Links

It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

scholarly journals Four-dimensional aspects of tight contact 3-manifolds

2021 ◽  
Vol 118 (22) ◽  
pp. e2025436118
Author(s):  
Matthew Hedden ◽  
Katherine Raoux
Keyword(s):  
Euler Characteristic ◽  
Contact Structure ◽  
Fiber Surface ◽  
Affirmative Answer ◽  
Contact Structures ◽  
Torus Knots ◽  
Tight Contact ◽  
Embedded Surfaces ◽  
Fibered Link

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Y×[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces in Y×[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant.

scholarly journals The Legendrian knot complement problem

2018 ◽  
Vol 27 (14) ◽  
pp. 1850067 ◽  
Author(s):  
Marc Kegel
Keyword(s):  
Contact Structure ◽  
Legendrian Knot ◽  
Tight Contact ◽  
Legendrian Links ◽  
User Friendly ◽  
The Way

We prove that every Legendrian knot in the tight contact structure of the [Formula: see text]-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight [Formula: see text]-sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given.

scholarly journals Fiberwise convexity of Hill’s lunar problem

2017 ◽  
Vol 09 (04) ◽  
pp. 571-630 ◽  
Author(s):  
Junyoung Lee
Keyword(s):  
Energy Level ◽  
Contact Structure ◽  
Finsler Geometry ◽  
Critical Energy ◽  
Critical Value ◽  
Finsler Metrics ◽  
Tight Contact ◽  
Systolic Inequality ◽  
Geodesic Problem ◽  
Hill's Lunar Problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

scholarly journals CONTACT STRUCTURES, σ-CONFOLIATIONS AND CONTAMINATIONS IN 3-MANIFOLDS

2009 ◽  
Vol 11 (02) ◽  
pp. 201-264 ◽  
Author(s):  
ULRICH OERTEL ◽  
JACEK ŚWIATKOWSKI
Keyword(s):  
Contact Structure ◽  
Contact Structures ◽  
Tight Contact ◽  
Branched Surface ◽  
Definition Of ◽  
Branched Surfaces ◽  
Natural Way

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures. Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

scholarly journals EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS

2006 ◽  
Vol 17 (09) ◽  
pp. 1013-1031 ◽  
Author(s):  
TOLGA ETGÜ ◽  
BURAK OZBAGCI
Keyword(s):  
Contact Structure ◽  
Complex Surface ◽  
Contact Structures ◽  
Open Book ◽  
Open Books ◽  
Tight Contact ◽  
Weinstein Conjecture ◽  
Circle Bundles ◽  
Surface Singularities

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

scholarly journals The diffeotopy group of S1×S2 via contact topology

2010 ◽  
Vol 146 (4) ◽  
pp. 1096-1112 ◽  
Author(s):  
Fan Ding ◽  
Hansjörg Geiges
Keyword(s):  
Fundamental Group ◽  
Rotation Number ◽  
Contact Structure ◽  
Alternative Proof ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Topology ◽  
Tight Contact ◽  
Contact Surgery

AbstractAs shown by Gluck in 1962, the diffeotopy group of S1×S2 is isomorphic to ℤ2⊕ℤ2 ⊕ℤ2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S1×S2, based at the standard tight contact structure, is isomorphic to ℤ; (ii) inspired by previous work of Fraser, an example is given of an integer family of Legendrian knots in S1×S2#S1×S2 (with its standard tight contact structure) that can be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston–Bennequin invariant, and rotation number).

scholarly journals Tight contact structures via admissible transverse surgery

2019 ◽  
Vol 28 (04) ◽  
pp. 1950032 ◽  
Author(s):  
J. Conway
Keyword(s):  
Contact Structure ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Tight Contact ◽  
Transverse Knots ◽  
Structure Formula ◽  
Heegaard Floer

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.

Geometry of Legendrian knots

2016 ◽  
Vol 25 (13) ◽  
pp. 1650069
Author(s):  
Dishant M. Pancholi ◽  
Suhas Pandit
Keyword(s):  
Contact Structure ◽  
Total Curvature ◽  
Legendrian Knots ◽  
Linking Number ◽  
Legendrian Knot ◽  
Tight Contact ◽  
Torus Knot ◽  
Extrinsic Geometry ◽  
Bounded Below ◽  
Explicit Relation

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on [Formula: see text] In particular, we show that the total curvature of a Legendrian knot [Formula: see text] in [Formula: see text] is bounded below by [Formula: see text] times, the total number of cusps in the front projection of [Formula: see text]. We also show that a Legendrian [Formula: see text]-torus knot has the total curvature bounded below by [Formula: see text] while that of the Legendrian knots [Formula: see text] is bounded below by [Formula: see text]. Furthermore, we find an explicit relation between the Thurston–Bennequin number of a Legendrian knot [Formula: see text] and the geometric self-linking number, the curvature and the torsion of the knot [Formula: see text].

scholarly journals Maximal Thurston–Bennequin number and reducible Legendrian surgery

2016 ◽  
Vol 152 (9) ◽  
pp. 1899-1914 ◽  
Author(s):  
Kouichi Yasui
Keyword(s):  
Contact Structure ◽  
Tight Contact

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$, and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

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