Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, motivated by 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.