Multi-triangulations as complexes of star polygons

A k-triangulation of a convex polygon is a maximal set of diagonals so that no k+1 of them mutually cross. k-triangulations have received attention in recent literature, with motivation coming from several interpretations of them.We present a new way of looking at k-triangulations, where certain star polygons naturally generalize triangles for k-triangulations. With this tool we give new, direct proofs of the fundamental properties of k-triangulations (number of edges, definition of flip). This interpretation also opens up new avenues of research that we briefly explore in the last section.