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 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).

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 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 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.

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.

Incoherent nullification of torus knots and links

2019 ◽  
Vol 28 (05) ◽  
pp. 1950033
Author(s):  
Zac Bettersworth ◽  
Claus Ernst
Keyword(s):  
Upper Bound ◽  
Torus Knots ◽  
Number Formula ◽  
Knots And Links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

Nullification of Torus knots and links

2014 ◽  
Vol 23 (11) ◽  
pp. 1450058 ◽  
Author(s):  
Claus Ernst ◽  
Anthony Montemayor
Keyword(s):  
Torus Knots ◽  
The Family ◽  
Minimum Number ◽  
Knots And Links

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

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 Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

2020 ◽  
Author(s):  
Michael O'Keeffe ◽  
Michael Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Torus Knots ◽  
Vertex Transitive ◽  
Knots And Links ◽  
Complete Account

<p>We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed <i>regular</i>. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.</p>

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