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 $d \geq 3$. This answers a question of Danaraj and Klee [10]; it also strengthens a result of Lickorish [16] 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.”