Diagonal Complexes for Punctured Polygons

Abstract

It is known that taken together, all collections of nonintersecting diagonals in a convex planar n-gon give rise to a (combinatorial type of a) convex (n − 3)-dimensional polytope As_n called the Stasheff polytope, or associahedron. In the paper, we act in a similar way by taking a convex planar n-gon with k labeled punctures. All collections of mutually nonintersecting and mutually nonhomotopic topological diagonals yield a complex As_n,k. We prove that it is a topological ball. We also show a natural cellular fibration Asn,k → As_n,k−1. A special example is delivered by the case k = 1. Here the vertices of the complex are labeled by all possible permutations together with all possible bracketings on n distinct entries. This hints to a relationship with M. Kapranov’s permutoassociahedron.