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 [8]; it also strengthens a result of Lickorish [13] 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.&quot