Hopf index properties

In this way, a complex function < )(r) can be interpreted as a map S3< S2. This is very important, since maps of this kind can be classified in homotopy classes labeled by a topological integer number called the Hopf index, so that the same topological property applies to any scalar field (provided that it is onevalued at infinity). 

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. 

Consider a smooth map/ S3 —> S2. We have called the fiber of a point p C S1 to the inverse image f l p), which is generally a closed curve in S3. Now we define the multiplicity of the fiber / 1(p) to the number of connected components of / (//. Consider the map f1 S3 — S2, where n is an integer, for/" to be a good smooth map. The linking number of the closed curves that form the fibers of /" is equal to the linking number of the closed curves that form the fibers off (they are the same curves). However, the multiplicity of the fibers of/" is equal to n times the multiplicity of the fibers of/. Consequently, the Hopf index has the following property ... 

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