  Maximal $(k+1)$-crossing-free graphs on a planar point set in convex
position, that is, $k$-triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
  We introduce a new way of looking at $k$-triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of $k$-triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.
