As has been pointed out to me, in the modern method of using analytical continuation on its own to get a Riemann surface a system of cuts is not used. But this is no good if one wants to study branches of functions. Cuts are boundaries of a domain for branches and if such boundaries are not defined the domain is not defined. Then without a domain the branches themselves are not defined apart from just their initial values. The full properties of branches cannot be got from such a minimal definition.
   So in order for branches to be fully defined a domain has to be provided for them. The use of cuts to produce a simply-connected region where the monodromy theorem applies is familiar and so single-valuedness of each branch on the domain is obtained. A branch is just a function from the cut plane to a region of the w-plane and is distinguished from its fellows by its initial value. When each branch-point is given a cut their combined images in the w-plane separate each initial value from every other. Each is in a range disjoint from the others. However when a node has more than one associated branch-point, which is the case when the defining relation is non- linear in the independent variable, it ts necessary to harmonize the cuts so that no redundant boundaries are created. Provide one of those branch-points with a cut. This defines boundary-elements which meet at the node. Take one sucb boundary-element and image it with cuts terminating in the remaining branch-points.
  Permutations on branches are intrinsic to closed circuits but one wants to associate them with crossings of cuts. NB: A permutation cannot generally be associated consistently with just a branch-point. Put an arrow (orientation counts) across each cut. Now take a circuit from and to the initial point which crosses one cut only in the direction of its arrow. The permutation produced by the circuit is given to the arrow. Crossing the opposite way gives the inverse permutation. Cuts from branch-points associated with the same node get the same permutation.
  A general circuit crosses more than one cut. The permutation it produces is the same as the product of the assigned permutations taken in the order in which the cuts are crossed. This is shown by distorting the circuit back to the initial point in between crossings,
  Now translatability, to get permutations for a system of cuts from those assumed already known for another system. A circuit crossing one cut only in the system for determination may cross more than one cut in the known system. But its permutation is determinable by the previous paragraph and this permutation just has to be given to the crossed cut.
  © Jan2004  and  Aug2010  revision.