Non-constructible Complexes and the Bridge Index

We show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds concluding that a 3-sphere or 3-ball is non-shellable or not vertex decomposable. These two last bounds are sharp.