Shelling Polyhedral 3-Balls and 4-Polytopes

There is a long history of constructions of non-shellable triangulations of -dimensional (topological) balls. This paper gives a survey of these constructions, including Furch’s 1924 construction using knotted curves, which also appears in Bing’s 1962 survey of combinatorial approaches to the Poincaré conjecture, Newman’s 1926 explicit example, and M. E. Rudin’s 1958 non-shellable triangulation of a tetrahedron with only vertices (all on the boundary) and facets. Here an (extremely simple) new example is presented: a non-shellable simplicial -dimensional ball with only vertices and facets. It is further shown that shellings of simplicial -balls and -polytopes can “get stuck”: simplicial -polytopes are not in general “extendably shellable.” Our constructions imply that a Delaunay triangulation algorithm of Beichl & Sullivan, which proceeds along a shelling of a Delaunay triangulation, can get stuck in the 3D version: for example, this may happen if the shelling follows a knotted curve.