Electromagnetism Hopf index

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. 

In an electromagnetic knot, each line is labeled by a complex number. If there are m lines with the same label, we will say that m is the multiplicity. If all the pairs of line have the same linking number l, it turns out that the Hopf index is given as n = Im2. 

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. 

It is easy to show that we can also construct electromagnetic knots with Hopf index — n2 by means of the dual fields... 

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