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The complex of non-crossing diagonals of a polygon. (English) Zbl 1196.52008
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.
MSC:
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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