scholarly journals Orbit growth of contact structures after surgery

2021 ◽  
Vol 4 ◽  
pp. 1103-1141
Author(s):  
Patrick Foulon ◽  
Boris Hasselblatt ◽  
Anne Vaugon
Keyword(s):  
Contact Structures

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 Contact structures and reducible surgeries

2015 ◽  
Vol 152 (1) ◽  
pp. 152-186 ◽  
Author(s):  
Tye Lidman ◽  
Steven Sivek
Keyword(s):  
Contact Structures ◽  
Surgery Theory ◽  
Contact Topology ◽  
Exceptional Surgery ◽  
Genus 2 ◽  
Cabling Conjecture

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

scholarly journals Microfluidic Optical Shutter Flexibly x-y Actuated via Electrowetting-on-Dielectrics with <20 ms Response Time

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Henning Fouckhardt ◽  
Johannes Strassner ◽  
Carina Heisel ◽  
Dominic Palm ◽  
Christoph Doering
Keyword(s):  
Optical Properties ◽  
Response Time ◽  
Visible Spectral Range ◽  
Contact Structures ◽  
Light Path ◽  
Cell Type ◽  
Electrically Conductive ◽  
Droplet Movement ◽  
In Series ◽  
Optical Shutter

Tunable microoptics deals with devices of which the optical properties can be changed during operation without mechanically moving solid parts. Often a droplet is actuated instead, and thus tunable microoptics is closely related to microfluidics. One such device/module/cell type is an optical shutter, which is moved in or out of the path of the light. In our case the transmitting part comprises a moving transparent and electrically conductive water droplet, embedded in a nonconductive blackened oil, that is, an opaque emulsion with attenuation of 30 dB at 570 nm wavelength over the 250 μm long light path inside the fluid (15 dB averaged over the visible spectral range). The insertion loss of the cell is 1.5 dB in the “open shutter” state. The actuation is achieved via electrowetting-on-dielectrics (EWOD) with rectangular AC voltage pulses of 2·90 V peak-to-peak at 1 kHz. To flexibly allow for horizontal, vertical, and diagonal droplet movement in the upright x-y plane, the contact structures are prepared such that four possible stationary droplet positions exist. The cell is configured as two capacitors in series (along the z axis), such that EWOD forces act symmetrically in the front and back of the 60 nl droplet with a response time of <20 ms.

scholarly journals Tight contact structures on some small Seifert fibered 3-manifolds

2007 ◽  
Vol 129 (5) ◽  
pp. 1403-1447 ◽  
Author(s):  
Paolo. Ghiggini ◽  
Paolo. Lisca ◽  
András. Stipsicz
Keyword(s):  
Contact Structures ◽  
Tight Contact

scholarly journals COMBED 3-MANIFOLDS WITH CONCAVE BOUNDARY, FRAMED LINKS, AND PSEUDO-LEGENDRIAN LINKS

2001 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Vector Field ◽  
Finite Order ◽  
Singular Vector ◽  
Contact Structures ◽  
Winding Number ◽  
Legendrian Links ◽  
Concave Boundary

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M, v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly ∂M≠∅, and v may be tangent to ∂M, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M, v, L), with (M, v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transverality-preserving simultaneous homotopy of v and isotopy of L). All realizations are based on the notion of branched standard spine, and build on results previously obtained, Links are encoded by means of diagrams on branched spines, where the diagram is C 1 with respect to the branching. Several motivations for being interested in combinatorial realizations of the topological categories considered in this paper are given in the introduction. The encoding of links is suitable for the comparison of the framed and the pseudo-Legendrian categories, and some applications are given in connection with contact structures, torsion and finite-order invariants. An estension of Trace's notion of winding number of a knot diagram is introduced and discussed.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

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