### 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 not shellable, but evasive. 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.

Original language | English (US) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 20 |

Issue number | 3 |

State | Published - Aug 30 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

- Collapsibility
- Discrete Morse theory
- Knots in triangulations
- Local constructibility
- Non-evasiveness
- Shellability

### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*20*(3).

**Knots in collapsible and non-collapsible balls.** / Benedetti, Bruno; Lutz, Frank H.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 20, no. 3.

}

TY - JOUR

T1 - Knots in collapsible and non-collapsible balls

AU - Benedetti, Bruno

AU - Lutz, Frank H.

PY - 2013/8/30

Y1 - 2013/8/30

N2 - We construct the first explicit example of a simplicial 3-ball B15,66 that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B12,38 with 12 vertices that is collapsible and not shellable, but evasive. Finally, we present the first explicit triangulation of a 3-sphere S18,125 (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.

AB - We construct the first explicit example of a simplicial 3-ball B15,66 that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B12,38 with 12 vertices that is collapsible and not shellable, but evasive. Finally, we present the first explicit triangulation of a 3-sphere S18,125 (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.

KW - Collapsibility

KW - Discrete Morse theory

KW - Knots in triangulations

KW - Local constructibility

KW - Non-evasiveness

KW - Shellability

UR - http://www.scopus.com/inward/record.url?scp=84883334193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883334193&partnerID=8YFLogxK

M3 - Article

VL - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

ER -