Interpolating curves by subdivision surfaces is one of the major constraints that is partially addressed in the literature. So far, no more than two intersecting curves can be interpolated by a subdivision surface such as Doo-Sabin or Catmull-Clark surfaces. One approach that has been used in both of theses surfaces is the polygonal complex approach where a curve can be defined by a control mesh rather than a control polygon. Such a definition allows a curve to carry with it cross derivative information which can be naturally embodied in the mesh of a subdivision surface. This paper extends the use of this approach to interpolate an unlimited number of curves meeting at an extraordinary point on a subdivision surface. At that point, the curves can all meet with eitherC0orC1continuity, yet still have common tangent plane. A straight forward application is the generation of subdivision surfaces through 3-regular meshes of curves for which an easy interface can be used.