The topology of inverse rational functions is investigated here in somewhat more detail than in the previous Note3. Given a defining equation
(1)   zf(w)-g(w)=0
the possibility of a bilinear transformation on the independent variable z which does not essentially change the number and nature of singularities allows it to be assumed that the degree, say n, of f is greater than the degree, say m, of g. It is assumed that the zeros of f and g are all different and can be taken to be variable non-vanishing parameters.
  Let r be a zero of f and require one zero of g to have a value (1+q^2)r where |q| is small. Then among the nodes there are two which differ from r by a term of the order of q. The calculations to establish this are left to the reader. Then by a construction similar to that used in Note3 a cut in the z-plane and a closed region in the w-plane are derived. It is required that r be inside the region and this can be ensured by giving an appropriate value to arg q which is at our disposal.
  Let this construction be carried out for m zeros of f and all the zeros of g. There are n+m-1 nodes with corresponding transposing branch-points and 2m of them are now disposed of. There are n branches and so far m are defined leaving n-m to be defined. Cuts from each of the remaining n-m-1 branch-points to 0 which is itself a branch-point of order n-m are imaged by boundaries going to infinity. Note that infinity is not a branch-point so there is no need for a cut to go there. Now there are as there should be n regions separated by n-1 boundaries.
  The initial point can be any point of the z-plane that is not a singularity. By the monodromy theorem no two initial values can be in the same region. Then since there are as many initial values as regions there must be one in each region as required.
  Permutations of branches across cuts can be read off from the adjacency of regions to the corresponding boundary.
  The topology of the system under continuous movement of the singularities and cuts is preserved as long as there are no collisions.

Copyright March 2008 conesetter.