Electromagnetic knots properties

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

The electromagnetic knot given in the previous subsections, a representative of the homotopy class C, can be easily generalized to classes C 2. To do that, we will need a property of the Hopf index. 

The electromagnetic knots satisfy a very important property. In a precise way, the following proposition holds tme. 

Proposition 2. Any standard radiation electromagnetic field in empty space with Faraday 2-form. F, regular in a bounded spacetime domain D, coincides locally with a knot around any point P C D in the following sense. There is a knot with 2-form 3Fkn, such that Fst = Fkn around P, except perhaps ifP is in a zero measure set. The same property holds for Fst. 

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