About homotopy classes of non-singular vector fields on the three-sphere

Dufraine, Emmanuel

doi:10.1007/bf02969412

Cited by 5 publications

(3 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where n, m, l and s are positive integer numbers. These fields are related to the Seifert fibrations [35]. The main difference is that, in equations (A.3), (A.4), the notation η n ( ) , η being a complex number, means to leave the modulus of η invariant while the phase of η is multiplied by n.…”

Section: Discussionmentioning

confidence: 99%

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

Arrayás¹,

Trueba²

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We present a new range of solutions of Maxwell's equations in vacuum in which the topology of the field lines is that of the whole torus knots set. Knotted electromagnetic fields are solutions of the Maxwell equations in vacuum in which magnetic lines, and also electric lines, have some kind of linkage. These solutions may play an important role in fundamental physics problems but also have practical interest and applications.

show abstract

Section: Discussionmentioning

confidence: 99%

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

Arrayás¹,

Trueba²

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…where n, m, l and s are positive integer numbers. These scalar fields are related to the Seifert fibrations [64]. The main difference is that, in equations (149) and (150)), the notation η (n) , η being a complex number, means to leave the modulus of η invariant while the phase of η is multiplied by n.…”

Section: Construction Of the Classmentioning

confidence: 99%

Knots in electromagnetism

2017

View full text Add to dashboard Cite

Maxwell equations in vacuum allow for solutions with a non-trivial topology in the electric and magnetic field line configurations at any given moment in time. One example is a space filling congruence of electric and magnetic field lines forming circles lying on the surfaces of nested tori. In this example the electric, magnetic and Poynting vector fields are orthogonal everywhere. As time evolves the electric and magnetic fields expand and deform without changing the topology and energy, while the Poynting vector structure remains unchanged while propagating with the speed of light. The topology is characterized by the concept of helicity of the field configuration. Helicity is an important fundamental concept and for massless fields it is a conserved quantity under conformal transformations. We will review several methods by which linked and knotted electromagnetic (spin-1) fields can be derived. A first method, introduced by A. Rañada, uses the formulation of the Maxwell equations in terms of differential forms combined with the Hopf map from the three-sphere S 3 to the two-sphere S 2. A second method is based on spinor and twistor theory developed by R. Penrose in which elementary twistor functions correspond to the family of electromagnetic torus knots. A third method uses the Bateman construction of generating null solutions from complex Euler potentials. And a fourth method uses special conformal transformations, in particular conformal inversion, to generate new linked and knotted field configurations from existing ones. This fourth method

show abstract

“…Any regular value of the maps φ 0 or θ 0 , has a 1-dimensional preimage in R 3 which depending on the integers n, m, l, s is a torus knot [19]. For the map φ 0 , the Hopf index of a pair of such preimages is H(φ 0 ) = nm and for the map θ 0 is H(θ 0 ) = ls.…”

Section: Initial Conditions For the Fieldsmentioning

confidence: 99%

On the Fibration Defined by the Field Lines of a Knotted Class of Electromagnetic Fields at a Particular Time

Arrayás¹,

Trueba²

2017

Symmetry

View full text Add to dashboard Cite

A class of vacuum electromagnetic fields in which the field lines are knotted curves are reviewed. The class is obtained from two complex functions at a particular instant t = 0 so they inherit the topological properties of red the level curves of these functions. We study the complete topological structure defined by the magnetic and electric field lines at t = 0. This structure is not conserved in time in general, although it is possible to red find special cases in which the field lines are topologically equivalent for every value of t.

show abstract

About homotopy classes of non-singular vector fields on the three-sphere

Cited by 5 publications

References 15 publications

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

Knots in electromagnetism

On the Fibration Defined by the Field Lines of a Knotted Class of Electromagnetic Fields at a Particular Time

Contact Info

Product

Resources

About