Legendrian Hopf Links

2020 ◽  
Vol 71 (4) ◽  
pp. 1419-1459
Author(s):  
Hansjörg Geiges ◽  
Sinem Onaran
Keyword(s):  
Contact Structure ◽  
Hopf Links ◽  
Hopf Link

Abstract We completely classify Legendrian realizations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

HOMFLY Polynomials of Some Generalized Hopf Links

2000 ◽  
Vol 09 (07) ◽  
pp. 865-883 ◽  
Author(s):  
TAT-HUNG CHAN
Keyword(s):  
Explicit Formulas ◽  
Recurrence Equations ◽  
Homfly Polynomials ◽  
Hopf Links ◽  
Hopf Link

The Hopf link, consisting of two unknots wrapped around each other, is the simplest possible nontrivial link with more than one component. We can generalize it to two bundles of "parallel" unknots wrapped around each other. In this paper, we show that when one of the two bundles has a fixed side, the HOMFLY polynomials of the links satisfy a system of recurrence equations. This leads to a procedure for computing explicit formulas for the HOMFLY polynomials.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

scholarly journals Quaternionic contact 4n + 3-manifolds and their 4n-quotients

2021 ◽  
Author(s):  
Yoshinobu Kamishima
Keyword(s):  
Scalar Curvature ◽  
Lie Group ◽  
Contact Structure ◽  
Einstein Manifolds ◽  
Hermitian Metrics ◽  
Kähler Metric ◽  
Formula One ◽  
Quaternion Space ◽  
Kahler Metric ◽  
Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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.

Dynamics of a Contact Structure Along a Vector Field of its Kernel

2008 ◽  
Vol 8 (1) ◽  
Author(s):  
Abbas Bahri ◽  
Yongzhong Xu
Keyword(s):  
Vector Field ◽  
Variational Problem ◽  
Contact Structure ◽  
Contact Form ◽  
Dual Form

AbstractIn this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).

scholarly journals Multi-symplectic formulation of near-local Hamiltonian balanced models

2011 ◽  
Vol 467 (2136) ◽  
pp. 3424-3442 ◽  
Author(s):  
Sylvain Delahaies ◽  
Peter E. Hydon
Keyword(s):  
Potential Vorticity ◽  
Contact Structure ◽  
Fluid Motion ◽  
Local Model ◽  
Contact Transformation ◽  
Geometric Features ◽  
Coriolis Parameter ◽  
New Class

We transform near-local Hamiltonian balanced models (HBMs) describing nearly geostrophic fluid motion (with constant Coriolis parameter) into multi-symplectic (MS) systems. This allows us to determine conservation of Lagrangian momentum, energy and potential vorticity for Salmon's L 1 dynamics; a similar approach works for other near-local balanced models (such as the -model). The MS approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model. The contact structure yields a contact transformation that makes the problem of front formation tractable. The new class includes the first local model with a variable Coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model.

Heat-Resistant Barrier Contacts Made on the Basis of TiBx and ZrBx to SiC and GaN

2009 ◽  
Vol 615-617 ◽  
pp. 565-568 ◽  
Author(s):  
Alexander A. Lebedev ◽  
A.E. Belyaev ◽  
N.S. Boltovets ◽  
V.N. Ivanov ◽  
Raisa V. Konakova ◽  
...  
Keyword(s):  
Silicon Carbide ◽  
Phase Composition ◽  
Heat Resistance ◽  
Contact Structure ◽  
Schottky Barriers ◽  
Electrophysical Properties ◽  
Depth Profiles ◽  
Contact Metallization ◽  
Before And After ◽  
Concentration Depth Profiles

We studied the heat resistance of AuTiBx (ZrBx) barrier contacts to n-SiC 6H and n-GaN. The Schottky barrier diode (SBD) parameters, the concentration depth profiles for contact structure components and the phase composition of contact metallization were measured both before and after rapid thermal annealing (RTA) at temperatures up to 900 °С (1000 °С) for contacts to GaN (SiC 6H). It is shown that the layered structure of metallization and electrophysical properties of Schottky barriers (SBs) remain stable after RTA, thus indicating their heat resistance. The ideality factor n of the I-V characteristic of SBDs after RTA was 1.2, while the SB height φВ was ~0.9 eV (~0.8 eV) for the gallium nitride (silicon carbide) barrier structures.

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