Non-evasiveness, collapsibility and explicit knotted triangulations

Non-evasiveness is a further strengthening of collapsibility, emerged in theoretical computer science and later studied by Kahn, Saks and Sturtevant [9] and Welker [11]. A 0-dimensional complex is non-evasive if and only if it consists of a single point. Recursively, a d-dimensional simplicial complex (d > 0) is non-evasive if and only if there is some vertex v whose link and deletion are both non-evasive. Every non-evasive complex is collapsible. The converse is false: Collapsibility is not maintained under taking links. In fact, there are elementary examples of collapsible 2-complexes all of whose vertex links are non-contractible. A first such example with only six vertices was found by Björner; for another example, see Barmak–Minian [1, Figure 7]. However, the difference between collapsibility and non-evasiveness does not simply depend on vertex links. In fact, we show that even a manifold can be collapsible and evasive: