Visibility Graphs and Deformations of Associahedra

The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P , we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P . We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P , a deformation space of polygons is created which encapsulates substructures of the associahedron. 1. Associahedra from Polygons 1.1. Let P be a simple planar polygon with labeled vertices. Unless mentioned otherwise, assume the vertices of P to be in general position, with no three collinear vertices. A diagonal of P is a line segment connecting two vertices of P which is contained in the interior of P . A diagonalization of P is a partition of P into smaller polygons using noncrossing diagonals of P . Let a convex diagonalization of P be one which divides P into smaller convex polygons. Definition 1. Let π(P ) be the poset of all convex diagonalizations of P where for a ≺ a if a is obtained from a by adding new diagonals. Figure 1 shows diagonalizations of a polygon P . Parts (b) through (d) show some elements of π(P ) where part (c) is greater than (d) in the poset ordering. ( a ) ( b ) ( c ) ( d ) Figure 1. A polygon with (a) a diagonalization and (b) – (d) convex diagonalizations. If P is a convex polygon, all of its diagonalizations will obviously be convex. The question was asked in combinatorics as to whether there exists a convex polytope whose face poset is isomorphic to π(P ) for a convex polygon P . It was independently proven in the affirmative by Lee [8] and Haiman (unpublished): 2000 Mathematics Subject Classification. Primary 52B11.