An equation f(w,z,p,q...)=0 together with the equation to 0 of the partial derivative with respect to w are two equations for pairs of nodes and branch-points. By the local theory they are thereby defined as differentiable functions or the parameters p,q... except at values which make either the first partial derivative with respect to z or the second partial derivative with respect to w vanish. Since differentiability implies continuity varying the parameters over allowed values gives a class of implicit functions with shared topology. So to get that topology the parameters may be given numerical values which allow Mahler's Principle of Laziness to be applied.
  An example: zg(w)-h(w)=0 where g is of degree 3 and h of degree 2. The chosen case is z(w^3-1)-3w^2=0. As z tends to infinity there are range limits at the cube roots of unity and as z tends to zero there is a range limit which is a node at zero and a range limit at infinity. There is a branch-point at 0 and there are three branch-points at minus the cube roots of 4. To define a set of branches take a cut from 0 to infinity along the positive real axis and for the other three take radial cuts. This gives one branch say P with an unbounded range and two say Q and R both with bounded ranges. The permutations on branches are (PR) across the negative real cut, (QR) across the positive real cut and (PQ) across the other two.
  An alternative system of cuts replaces the cuts which have the permutation (PQ) with one cut with that permutation. Then the other two cuts both get the permutation (QR). One can go on to replace these two cuts with one as well.
  For anyone who finds this obscure it is recommended to draw some pictures.
    
  Copyright Nov06 conesetter