scholarly journals On Bennequin-type inequalities for links in tight contact 3-manifolds

2020 ◽  
Vol 29 (08) ◽  
pp. 2050055
Author(s):  
Alberto Cavallo
Keyword(s):  
Tubular Neighborhood ◽  
The Self ◽  
Rational Homology ◽  
Linking Number ◽  
Tight Contact ◽  
Legendrian Links ◽  
Thurston Norm ◽  
Similar Inequality

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere [Formula: see text], whenever [Formula: see text] is tight. More specifically, we show that the self-linking number of a transverse link [Formula: see text] in [Formula: see text], such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm [Formula: see text] of [Formula: see text]. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-positive link in [Formula: see text]. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.

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 AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

2020 ◽  
pp. 1-33
Author(s):  
ALBERTO CAVALLO
Keyword(s):  
Chain Complex ◽  
Open Book ◽  
Rational Homology ◽  
Open Book Decompositions ◽  
Chain Complexes ◽  
Legendrian Links ◽  
Sum Formula ◽  
A Chain ◽  
Special Cycle ◽  
Rational Homology Sphere

Abstract We introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES

2000 ◽  
Vol 09 (05) ◽  
pp. 703-711
Author(s):  
CARL J. STITZ
Keyword(s):  
Combinatorial Approach ◽  
Rational Homology ◽  
Characteristic Curves ◽  
Combinatorial Description ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Linking Numbers ◽  
Heegaard Diagram ◽  
Rational Homology Spheres ◽  
Matrix Invariants

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

scholarly journals Quaternions and hybrid nematic disclinations

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130204 ◽  
Author(s):  
Simon Čopar ◽  
Slobodan Žumer
Keyword(s):  
Liquid Crystals ◽  
Homotopy Theory ◽  
Classification Scheme ◽  
Colloidal Suspensions ◽  
The Self ◽  
Topological Classification ◽  
Winding Number ◽  
Linking Number ◽  
Disclination Line ◽  
Geometric Decomposition

Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number . The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.

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 LINKING NUMBER IN A PROJECTIVE SPACE AS THE DEGREE OF A MAP

2007 ◽  
Vol 16 (04) ◽  
pp. 489-497 ◽  
Author(s):  
JULIA VIRO DROBOTUKHINA
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
The Self ◽  
Real Projective Space ◽  
3 Dimensional ◽  
Linking Number ◽  
Degree Of A Map ◽  
Number Of Cycles ◽  
Dimensional Projective Space

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS

2005 ◽  
Vol 14 (06) ◽  
pp. 791-818 ◽  
Author(s):  
VLADIMIR CHERNOV TCHERNOV
Keyword(s):  
The Self ◽  
Compact Manifolds ◽  
Not Given ◽  
Connected Sum ◽  
Linking Number ◽  
Fundamental Fact ◽  
Linking Numbers

The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.

THE SELF-LINKING NUMBER OF A CLOSED CURVE IN ℝn

2000 ◽  
Vol 09 (04) ◽  
pp. 491-503
Author(s):  
A. MONTESINOS AMILIBIA ◽  
J. J. NUÑO BALLESTEROS
Keyword(s):  
Vector Bundle ◽  
Closed Curve ◽  
The Self ◽  
Regularity Conditions ◽  
Dimensional Vector ◽  
3 Dimensional ◽  
Orthogonal Vector ◽  
Linking Number

We introduce the self-linking number of a smooth closed curve α:S1→ℝn with respect to a 3-dimensional vector bundle over the curve, provided that some regularity conditions are satisfied. When n=3, this construction gives the classical self-linking number of a closed embedded curve with non-vanishing curvature [5]. We also look at some interesting particular cases, which correspond to the osculating or the orthogonal vector bundle of the curve.

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].

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