Elementary Moves on Polygonal Triangulations of the Disk

Definition 3 (Polygonal Triangulation). Let K be a two-dimensional simplicial complex (i.e., a collection of triangles). If K is homeomorphic to a disk, then it is called a triangulation of a disk. Furthermore, if K is a k-gon divided by diagonals into k − 2 triangles, then K will be called a polygonal triangulation. See Figure 1. A k-gon dissected into k − 2 triangles will be called a k-triangulation for the purposes of this paper. Note that a k-triangulation has k−3 diagonals. The vertices of a triangulation K will be enumerated v1, . . . , vk, moving clockwise around the outside of the k-gon.