scholarly journals Are Maxwell knots integrable?

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
A. Morozov ◽  
N. Tselousov
Keyword(s):  
Limit Cycles ◽  
Maxwell Equations ◽  
Poynting Vector ◽  
Measure Zero ◽  
Finite Energy ◽  
Link Invariant ◽  
Integral Of Motion ◽  
Integrable Structure ◽  
Field Lines ◽  
Knots And Links

AbstractWe review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. We also consider particular examples of the field lines for the particular family of finite energy source-free “knot” solutions, attempting to understand when the field lines are closed – and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada’s solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist.

scholarly journals Null Electromagnetic Fields from Dilatation and Rotation Transformations of the Hopfion

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1105 ◽  
Author(s):  
Manuel Arrayás ◽  
Antonio F. Rañada ◽  
Alfredo Tiemblo ◽  
José L. Trueba
Keyword(s):  
Electromagnetic Fields ◽  
Maxwell Equations ◽  
Symmetry Transformations ◽  
Field Lines

The application of topology concepts to Maxwell equations has led to the developing of the whole area of electromagnetic knots. In this paper, we apply some symmetry transformations to a particular electromagnetic knot, the hopfion field, to get a new set of knotted solutions with the properties of being null. The new fields are obtained by a homothetic transformation (dilatation) and a rotation of the hopfion, and we study the constraints that the transformations must fulfill in order to generate valid electromagnetic fields propagating in a vacuum. We make use of the Bateman construction and calculate the four-potentials and the electromagnetic helicities. It is observed that the topology of the field lines does not seem to be conserved as it is for the hopfion.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

ON THE G2 LINK INVARIANT

1993 ◽  
Vol 02 (04) ◽  
pp. 431-451 ◽  
Author(s):  
EFSTRATIA KALFAGIANNI
Keyword(s):  
Lie Algebra ◽  
Link Invariant ◽  
Markov Trace ◽  
Knots And Links

We construct a polynomial link invariant as Markov trace on certain one parameter algebras and we prove that it is equal to the invariant corresponding to the exeptional Lie algebra of type G2. We use braid representatives to calculate the invariant for several knots and links.

A LINK INVARIANT DOMINATING THE HOMFLY AND THE KAUFFMAN POLYNOMIALS

2010 ◽  
Vol 19 (11) ◽  
pp. 1507-1533
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Polynomial Invariant ◽  
Link Invariant ◽  
Knots And Links

By using a graph diagram named a magnetic graph diagram, we construct a polynomial invariant for knots and links. We show that it is a generalization of both the HOMFLY and the Kauffman polynomials.

Determination of two-dimensional magnetostatic equilibria and analogous Euler flows

1993 ◽  
Vol 246 ◽  
pp. 569-591 ◽  
Author(s):  
D. Linardatos
Keyword(s):  
Variational Problem ◽  
Energy State ◽  
Local Equilibrium ◽  
Minimum Energy ◽  
Saddle Points ◽  
Infinite Family ◽  
Finite Energy ◽  
Equilibrium Solutions ◽  
Energy Field ◽  
Field Lines

The equivalence of the method of magnetic relaxation to a variational problem with an infinity of constraints is established. This variational problem is solved in principle and approximations to the exact solution are compared to results obtained by numerical relaxation of fields with a single stationary elliptic point. In the case of a finite energy field of the above topology extending to infinity, we show that the minimum energy state is the one in which all field lines are concentric circles and that this state is topologically accessible from the original one. This state is used as a reference state for understanding the relaxation of fields constrained by finite boundaries. We then consider the relaxation of fields containing saddle points and confirm the tendency of the saddle points to collapse and form two Y-points. An infinite family of local equilibrium solutions each describing a Y-point is provided.

TM Standing Wave Solution of Maxwell Equations in a Self-Defocusing Kerr Media and its Application to a Thin-Film Waveguide

2021 ◽  
Vol 323 ◽  
pp. 100-112
Author(s):  
Ochirbat Nyamsuren ◽  
Purevdorj Munkhbaatar ◽  
Duger Ulam-Orgikh ◽  
Jav Davaasambuu ◽  
G. Ochirbat
Keyword(s):  
Thin Film ◽  
Film Thickness ◽  
Boundary Conditions ◽  
Standing Wave ◽  
Dielectric Function ◽  
Maxwell Equations ◽  
Function Method ◽  
Kerr Nonlinearity ◽  
Integral Of Motion ◽  
Kerr Media

We applied the dielectric function method to solve analytically L-NL-L structure problems with negative Kerr nonlinearity. A damped wave in linear and a periodic standing wave in non-linear media had to be matched at boundaries. We gave a formulation of boundary conditions that did not explicitly include a film thickness. The boundary-value of a dielectric function can be expressed through the constant of non-trivial integral of motion. Using it, one generates a family of matched solutions satisfying boundary conditions. Then arbitrary film thickness can be checked against this family of solutions in search of matches. As a result, all fitted solutions are determined straightforwardly.

scholarly journals Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

2018 ◽  
Vol 2020 (1) ◽  
pp. 214-286 ◽  
Author(s):  
Maria Chlouveraki ◽  
Jesús Juyumaya ◽  
Konstantinos Karvounis ◽  
Sofia Lambropoulou
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Type A ◽  
Closed Formula ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Linking Numbers ◽  
Markov Trace ◽  
Knots And Links

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

Untersuchung über die Ausbreitung von elektromagnetischen Wellen in einem zylindrischen Plasma mit Magnetfeld

1967 ◽  
Vol 22 (10) ◽  
pp. 1592-1599 ◽  
Author(s):  
Karl Weinhardt
Keyword(s):  
Maxwell Equations ◽  
Plasma Frequency ◽  
Arc Discharge ◽  
Collisionless Plasma ◽  
Electron Plasma ◽  
Coupling System ◽  
Field Lines ◽  
Cathode Arc ◽  
Electron Gyrofrequency ◽  
The Magnetic Field

Propagation of circularly symmetric electromagnetic modes parallel to the magnetic field lines in the positive column of an argon hollow-cathode arc discharge has been studied. The applied frequency (3·109 cps) was less than both the electron gyrofrequency and the electron plasma frequency. These measurements were compared with dispersion relations for circulary symmetric modes calculated by using the complete MAXWELL equations, the ε-tensor for a cold collisionless plasma, and suitable boundary conditions. It could be shown that the mode which was excited was most likely determined by the boundary of the coupling system and not by the boundary of the whole vessel as originally expected.

scholarly journals ULF Activity in the Earth Environment: Penetration of Electric Field from the Near-Ground Source to the Ionosphere under Different Configurations of the Geomagnetic Field

Atmosphere ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 801
Author(s):  
Vsevolod Yutsis ◽  
Yuriy Rapoport ◽  
Volodymyr Grimalsky ◽  
Asen Grytsai ◽  
Vasyl Ivchenko ◽  
...  
Keyword(s):  
Open Field ◽  
Electric Fields ◽  
Geomagnetic Field ◽  
Maxwell Equations ◽  
Closed Field ◽  
Dynamic Simulations ◽  
Ground Source ◽  
First Case ◽  
Earth Environment ◽  
Field Lines

The problem with the penetration of electric fields from atmospheric near-Earth electric current sources to the ionosphere is investigated both within the dynamic simulations of the Maxwell equations in the frequency domain and within the simplified quasi-electrostatic approach. Two cases of the geomagnetic field lines are considered. The first case is the penetration of the geomagnetic field lines deeply into the magnetosphere (open field lines), whereas the second one is the return of these lines into the Earth’s surface (closed field lines). The proper boundary conditions are formulated. It is demonstrated that in the case of the open field lines the results of the dynamic simulations differ essentially from the quasi-electrostatic approach, which is not valid there. In the case of the closed field lines, the results of simulations are practically the same both within the dynamic approach and within the quasi-electrostatic one. From realistic values of the densities of atmospheric electric currents ~0.1 µA/m2, the values of the electric fields within the ionosphere F-layer may reach about 1–10 mV/m.

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