Decompositions of simplicial balls and spheres with knots consisting of few edges

Abstract

Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every d >= 3. This answers a question of Danaraj & Klee it also strengthens a result of Lickorish about non-shellable spheres.

Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with ``few edges'' in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of ``few edges.''