## A vertex decomposable 3-ball with a knotted spanning arc consisting of 4 edges

Electronic Geometry Model No. 2001.05.004#### Author

Nikolaus Witte#### Description

Hachimori and Ziegler [1] proved that a
triangulation of a 3-ball which contains a knotted spanning arc
consisting of at most 3 edges is not vertex decomposable. The
following example (constructed along the ideas of Hachimori and
Ziegler [1], yet considerably smaller) shows that 3 is the best
possible upper bound for the number of edges.

Let *C*_{1} be the following
triangulation of a pile of 3*2*1 cubes: Each cube is split into 6
tetrahedra, and the tetrahedra are all grouped around one of the
diagonal axes of the cube. Furthermore let *a* denote the pink,
*b* the red, *c* the orange and *d* the light blue
vertex. For a face *F* and a vertex *x* let
*x***F* denote the cone over *F* with apex
*x*. Then *C* := *C*_{1} union (*v**(blue
faces)), where *v* is a vertex not contained in
*C*_{1}, is a 3-ball with the knotted spanning arc
*ab-bv-vc-cd*. Since we add *v**(blue faces) to
*C*_{1}, the edges *bv* and *vc* actually
exist. The edges *ab* and *cd* are already contained
in *C*_{1}. To visualize the knotted arc, one may
imagine it as following the yellow line. *C* is vertex decomposable: One can take
*v* as the first shedding vertex.

*Figure 1* depicts the pile of cubes and the
faces needed for the construction are colored blue. The yellow line
(including the yellow vertex *v*) visualizes the knotted
arc. To obtain the example
(*vert_dec_3-ball_with_knotted_arc.jvx*) one has to compleat the
construction by adding *v**(blue faces).

The master file *vert_dec_3-ball_with_knotted_arc.jvx* contains two
geometries: The facets of the 3-ball (encoded as their vertex sets)
and the knotted spanning arc (encoded as the vertex sets of its edges). In the
master file point *a* corresponds to 15, *b*
to 14, *c* to 9, *d* to 8 and *v* to 24. (If needed,
check the applet file for the indices of the other vertices.) As shedding
vertices one may use 24, 22, 23, 20, 18, 19, 21, 16, 17, 14, 12, 13, 15, 10, 4, 5, 11,
9, 0, 1, 3, 2, 6, 7 and 8 in this order.

**Keywords** | | Shellability; Constructibility; Simplicial Ball |

**MSC-2000 Classification** | | 52B22 (52B10, 57-99) |

**Zentralblatt No.** | | 05264880 |

#### References

- M. Hachimori and G. M. Ziegler:
*Decomposition of simplicial balls and spheres with knots consisting of few edges*, Mathematische Zeitschrift **235** (2000), 159-171.

#### Files

#### Submission information

Submitted: Tue May 8 20:14:13 CEST 2001.

Revised: Fri Jun 14 18:24:12 CEST 2002.

Accepted: Wed Nov 20 15:58:01 CET 2002.

#### Author's Address

Nikolaus Witte
TU Berlin

Fakultät 2, Institut für Mathematik, MA 6-2

Strasse des 17. Juni 136

10623 Berlin, Germany

witte@math.tu-berlin.de