## A vertex decomposable 3-ball and 3-sphere with a knot consisting of 6 edges

Electronic Geometry Model No. 2001.05.003#### Author

Nikolaus Witte#### Description

Hachimori and Ziegler [1] proved that a
triangulation of a 3-ball which contains a non-trivial knot consisting
of at most 5 edges is not vertex decomposable. The following example
(constructed along the ideas of Hachimori and Ziegler [1], yet
considerably smaller) shows that 5 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 the 3-ball *C*_{2} := *C*_{1}
union (*v**(blue faces)) union (*w**(green faces)), for
vertices *v*, *w* not contained in *C*_{1}, has
a non-trivial knot *ab-bv-vc-cd-dw-wa* with 6 edges. Since we add
*v**(blue faces), the edges *bv* and *vc* actually
exist and adding *w**(green faces) creates the edges *dw*
and *wa*. The edges *ab* and *cd* are already
contained in *C*_{1}. To visualize the knot, one may imagine it as
following the yellow line. *C*_{2} is vertex decomposable:
one can take *v* and *w* as the first two shedding
vertices.

In order to get a vertex decomposable 3-sphere with a
knot consisting of 6
edges, one takes the cone over the boundary of *C*_{2},
that is *C* := *C*_{2} union
(*u**boundary(*C*_{2})), *u* not contained in
*C*_{2}. The shelling of *C*_{2} can be
trivially extended to a shelling of *C* because
boundary(*C*_{2}) is a 2-sphere and therefore shellable.

*Figure 1* depicts the pile of cubes and the
faces needed for the construction are colored green and blue. The
yellow line (including the yellow vertices *v* and *w*)
visualizes the knot. To obtain the
example (*shel_3-ball_with_knot.jvx*) one has to compleat the
construction by adding *a**(green faces) and *b**(blue faces).

The master file *vert_dec_3-ball_with_knot.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, *v* to 24 and *w* to 25. (If needed,
check the applet file for the indices of the other vertices.) As shedding
vertices one may use 25, 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.** | | 05264879 |

#### 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 18:26:00 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