To create a system of cuts one may often start with a Mittag-Leffler star, a system of cuts where each finite branch-point gets its own cut going off to infinity. But infinity may not be a branch-point and then there is no need for a cut to go there. A system of cuts where each one has a branch-point at either end (including infinity if it is a branch-point) will be called standard. In a standard system  where there are n  branch-points  there can  be at most  n-1  cuts  as with more the connectivity of the plane would be lost.
 The simplest example is w² - wz + 1 = 0. The nodes are at +1 and -1 and the corresponding branch-points af +2 and -2. Cuts from fhem rsspectively to +infinity and -infinity along the real line, a star system, are together imaged by the two-part boundary consisting of the real line with 0 excluded. The ranges of the two branches are the upper and lower half-planes with 0 excluded. Going to a standard system we are limited to one cut which can be the part of the real line from -2 to +2. This is imaged  by a closed oval  through the nodes. The interior of the oval is the range of one branch, the exterior is the range of the other.
 Note how z is imaged for large |z|. In the standard system it is imaged in the bounded range near 0 and in the unbounded one near infinity. In the star system the imaging in each range is split between being near 0 and near infinity depending on arg z.
 Note also that the connectivities of the ranges are different in the star and standard systems. It  is the presence of  the range limit (or inverse pole as we also call it) that makes different connectivities possible.