scholarly journals Closed braids and knot holders associated to some laser dynamical systems: A pump-modulated Nd-doped Â…ber laser

2017 ◽  
Vol 12 (12) ◽  
pp. 6894-6900
Author(s):  
Elsaued Elrifai
Keyword(s):  
Dynamical Systems ◽  
Fiber Laser ◽  
Crossing Number ◽  
Topological Invariants ◽  
Control Parameters ◽  
Linking Number ◽  
Fibered Knots ◽  
Braid Theory ◽  
Knots And Links

In this work the arising knots and links for the pump-modulated Nd-doped fiber laser is investigated. For the associated templates, some of their topological invariants, such as braid linking matrix, braid words, crossing number and linking number, are studied. It is recognized that the derived topological invariants are quietly dependent on the control parameters Using the tools of the braid theory, it is shown that pump-modulated Nd-doped fiber laser knots and links are positive fibered knots and links.

GENERALIZED TEMPLATE FOR PERIOD DOUBLINGS

1995 ◽  
Vol 09 (18) ◽  
pp. 1123-1131
Author(s):  
T. ARIMITSU ◽  
T. MOTOIKE
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Crossing Number ◽  
Topological Characterization ◽  
Linking Number ◽  
Torus Knot ◽  
Knowing That ◽  
Parametric Pendulum

It is shown how one can derive the formula for the local crossing number with the help of the symbolic dynamics with two letters knowing that, on the extended (ξ, η)-template, any period-doubling sequences in continuous dynamical systems is characterized by the periodic block [Formula: see text] The formulas of the global crossing number and of the linking number are derived in terms of the local crossing number which can be extracted from the power spectrum of periodic orbits. The topological characterization of periodic orbits in continuous dynamical systems has come to be at hand for experimenters as well as for theorists. The formulas are applied to the situations of the numerical experiments for particular systems, and the results are investigated in connection with the concept of the irreducible and reducible templates. It is also shown how the formulas can be applicable to the forced Lorenz system, the Brusselator equation, the parametric pendulum, T(2, 5) resonant torus knot and P(7, 3, −2) pretzel knot.

Stick numbers of Montesinos knots

2021 ◽  
pp. 2150013
Author(s):  
Hwa Jeong Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Number Formula ◽  
Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (05) ◽  
pp. 625-649 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Weighted Sum ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

A Mechanical Filter Concept to Suppress Crane Load Oscillations

1997 ◽  
Author(s):  
B. Balachandran ◽  
Y.-Y. Li
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Control Parameters ◽  
Pivot Point ◽  
Preliminary Results ◽  
Scalar Control ◽  
Load Response ◽  
Mechanical Filter ◽  
Circular Track ◽  
Ship Roll

Abstract In this article, preliminary results obtained in the exploration of a mechanical filter concept for suppressing crane-load oscillations on a ship vessel are presented. The pivot point about which the load oscillates is constrained to follow a circular track in the considered filter. The governing dynamical systems for the cases with and without the filter are presented, and the nonlinear dynamics of these systems is studied with respect to quasi-static variation of different scalar control parameters. It is shown that the presence of the filter helps in eliminating some of the sub-critical bifurcations that may arise in the crane-load response during periodic ship-roll excitations.

Persistent Homology, Regimes and Climate Data

2021 ◽  
Author(s):  
Kristian Strommen ◽  
Nina Otter ◽  
Matthew Chantry ◽  
Joshua Dorrington
Keyword(s):  
Dynamical Systems ◽  
State Of The Art ◽  
Persistent Homology ◽  
Topological Invariants ◽  
Climate Data ◽  
Future Directions ◽  
Atlantic Sector ◽  
Linear Dynamical Systems ◽  
Markovian Dynamics ◽  
Shed Light

<p>The concept of weather or climate 'regimes' have been studied since the 70s, to a large extent because of the possibility they offer of truncating complicated dynamics to vastly simpler, Markovian, dynamics. Despite their attraction, detecting them in data is often problematic, and a unified definition remains nebulous. We argue that the crucial common feature across different dynamical systems with regimes is the non-trivial topology of the underlying phase space. Such non-trivial topology can be detected in a robust and explicit manner using persistent homology, a powerful new tool to compute topological invariants in arbitrary datasets. We show some state of the art examples of the application of persistent homology to various non-linear dynamical systems, including real-world climate data, and show how these techniques can shed light on questions such as how many regimes there really are in e.g. the Euro-Atlantic sector. Future directions are also discussed.</p>

Next step, synergetics?

2001 ◽  
Vol 24 (1) ◽  
pp. 66-67
Author(s):  
Wolfgang Tschacher ◽  
Ulrich M. Junghan
Keyword(s):  
Field Theory ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Organization Theory ◽  
Field Model ◽  
Environmental Control ◽  
Clinical Applications ◽  
Self Organization ◽  
Control Parameters ◽  
Cognitive Problem

Thelen et al. offer an inspiring behavior-based theory of a long-standing cognitive problem. They demonstrate how joining traditions, old (the Gestaltist field theory) and new (dynamical systems theory) may open up the path towards embodied cognition. We discuss possible next steps. Self-organization theory (synergetics) could be used to address the formation of gaze/reach attractors and their optimality, given environmental control parameters. Finally, some clinical applications of the field model are advocated.

scholarly journals Triple-crossing projections, moves on knots and links and their minimal diagrams

2020 ◽  
Vol 29 (04) ◽  
pp. 2050015 ◽  
Author(s):  
Michał Jabłonowski ◽  
Łukasz Trojanowski
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Systematic Method ◽  
Generating Set ◽  
New Type ◽  
Knots And Links ◽  
Minimal Generating Set

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

scholarly journals AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1369-1422 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Finite Number ◽  
Crossing Number ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Polynomial Coefficients ◽  
Bracket Polynomial ◽  
Knots And Links

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

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