an:05768626
Zbl 1196.52008
Braun, Benjamin; Ehrenborg, Richard
The complex of non-crossing diagonals of a polygon
EN
J. Comb. Theory, Ser. A 117, No. 6, 642-649 (2010).
00263862
2010
j
52B05
non-convex polygon; associahedra; simplicial complex; discrete Morse theory
Summary: Given a convex \(n\)-gon \(P\) in \(\mathbb R^2\) with vertices in general position, it is well known that the simplicial complex \(\theta (P)\) with vertex set given by diagonals in \(P\) and facets given by triangulations of \(P\) is the boundary complex of a polytope of dimension \(n - 3\). We prove that for any non-convex polygonal region \(P\) with \(n\) vertices and \(h+1\) boundary components, \(\theta (P)\) is a ball of dimension \(n+3h - 4\). We also provide a new proof that \(\theta (P)\) is a sphere when \(P\) is convex with vertices in general position.