Knots in collapsible and non-collapsible balls

TL;DR

This paper constructs explicit examples of complex 3-dimensional topological structures with specific properties, revealing new insights into the relationships between collapsibility, shellability, and knotted subcomplexes in simplicial complexes.

Contribution

It provides the first explicit examples of non-collapsible 3-balls, a non-shellable collapsible 3-ball, and a non-locally constructible 3-sphere, illustrating the impact of knotted subcomplexes on these properties.

Findings

Constructed a non-collapsible 3-ball with 15 vertices.

Presented a collapsible, evasive but non-shellable 3-ball with 12 vertices.

Explicitly triangulated a non-locally constructible 3-sphere with 18 vertices.

Abstract

We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 18 vertices) that is not locally constructible. All these examples are based on knotted subcomplexes with only three edges; the knots are the trefoil, the double trefoil, and the triple trefoil, respectively. The more complicated the knot is, the more distant the triangulation is from being polytopal, collapsible, etc. Further consequences of our work are: (1) Unshellable 3-spheres may have vertex-decomposable barycentric subdivisions. (This shows the strictness of an implication proven by Billera and Provan.) (2) For d-balls, vertex-decomposable implies non-evasive implies collapsible, and for d=3 all implications are strict. (This answers a question by Barmak.) (3) Locally constructible 3-balls may contain a double trefoil knot as a 3-edge subcomplex. (This improves a result of Benedetti and Ziegler.) (4) Rudin's ball is non-evasive.