In a system of cuts used to construct a simply-connected domain it may be that when the permutations on branches are calculated one or more cuts get the null permutation. Then those cuts as they have no effect can be omitted to leave a multiply-connected domain.
  An example: w^3 - 3w + 2z^2 = 0. The finite branch-points are at +1, -1, +i, -i and an obvious system of four cuts follows the axes to infinity. From this the two imaginary cuts can be replaced by one from +i to -i. A reader who has followed previous theory will be able to get the associated permutations.
  Different systems of cuts give the branches different ranges. But branches are identified by their initial values which are not changed so they retain their identities.
  It is also possible for ranges to be not simply-connected. Some examples are treated in the Notes.
  Copyright April 2010 conesetter.