scholarly journals Odd-symplectic forms via surgery and minimality in symplectic dynamics

2018 ◽  
Vol 40 (3) ◽  
pp. 699-713 ◽  
Author(s):  
HANSJÖRG GEIGES ◽  
KAI ZEHMISCH
Keyword(s):  
Contact Structure ◽  
Infinite Family ◽  
Hamiltonian Structures ◽  
Symplectic Forms ◽  
Symplectic Cobordism ◽  
Symplectic Dynamics ◽  
Standard Contact

We construct an infinite family of odd-symplectic forms (also known as Hamiltonian structures) on the $3$-sphere $S^{3}$ that do not admit a symplectic cobordism to the standard contact structure on $S^{3}$. This answers in the negative a question raised by Joel Fish motivated by the search for minimal characteristic flows.

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.

On 𝔬𝔰𝔭(1|2)-relative cohomology of the Lie superalgebra of contact vector fields on ℝ1|1

2017 ◽  
Vol 14 (02) ◽  
pp. 1750022
Author(s):  
Ben Fraj Nizar ◽  
Meher Abdaoui ◽  
Raouafi Hamza
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Contact Structure ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Differential ◽  
Standard Contact

We consider the [Formula: see text]-dimensional real superspace [Formula: see text] endowed with its standard contact structure defined by the 1-form [Formula: see text]. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra [Formula: see text]. We classify [Formula: see text]-invariant linear differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text], where [Formula: see text] is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the first differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text]. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on [Formula: see text] acting on bilinear differential operators, Int. J. Geom. Methods Mod. Phys. 2(1) (2005) 23–40].

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.

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 A Symplectic Dynamics Proof of the Degree–Genus Formula

2021 ◽  
Author(s):  
Peter Albers ◽  
Hansjörg Geiges ◽  
Kai Zehmisch
Keyword(s):  
Field Theory ◽  
Hopf Fibration ◽  
Contact Form ◽  
Projective Curves ◽  
Symplectic Field Theory ◽  
Genus Formula ◽  
Symplectic Dynamics ◽  
Standard Contact ◽  
Complex Projective

AbstractWe classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

scholarly journals On Subcritically Stein Fillable 5-manifolds

2018 ◽  
Vol 61 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Fan Ding ◽  
Hansjörg Geiges ◽  
Guangjian Zhang
Keyword(s):  
Fundamental Group ◽  
Contact Structure ◽  
Contact Structures ◽  
Homotopy Classification ◽  
Contact Manifolds ◽  
Almost Contact Structure ◽  
Diffeomorphism Type ◽  
Standard Contact ◽  
Simply Connected

AbstractWe make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

scholarly journals Integrable Evolutions of Twisted Polygons in Centro-Affine ℝm

2020 ◽  
Author(s):  
Gloria Marí Beffa ◽  
Annalisa Calini
Keyword(s):  
Integrable Systems ◽  
Affine Space ◽  
Arc Length ◽  
Hamiltonian Structures ◽  
Symplectic Forms ◽  
Geometric Invariants

Abstract We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler–Gel’fand–Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structures defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

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].

Non-Thrombogenicity of Clean Glass Revealed by Native Whole Blood Assay in Bead Columns

1983 ◽  
Vol 50 (04) ◽  
pp. 814-820 ◽  
Author(s):  
J A Bergeron ◽  
J M DiNovo ◽  
A F Razzano ◽  
W J Dodds
Keyword(s):  
Whole Blood ◽  
Glass Bead ◽  
Factor Xii ◽  
Platelet Factor ◽  
Serotonin Content ◽  
Whole Blood Assay ◽  
Vascular Integrity ◽  
Blood Assay ◽  
Standard Contact ◽  
Recalcification Time

SummaryThe previously described native whole blood assay for materials in solution or suspension has been adapted to materials in a bead column configuration. These experiments showed that the glass itself accounts for little or none of the high blood-reactivity observed with conventional glass bead columns. Columns composed solely of soft glass that was “cleaned” by heat treatment (500-595° C 18 hr, electric oven) were benign toward flowing native whole blood for all variables measured (platelet count and platelet-free plasma [C14]-serotonin content, platelet factor 3 and factor XII activities, and recalcification time) with the standard contact protocol. In addition, the effluent successfully maintained perfusion of the isolated kidney, a measure of the ability of platelets to support vascular integrity. Prolonged (30 min) normothermic contact with titrated whole blood increased the subsequent reactivity of initially clean glass toward whole blood albeit to a level much less than that of conventional glass bead columns.

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