## Furch's Ball

Electronic Geometry Model No. 2001.05.005#### Author

Nikolaus Witte#### Description

Furch [1] proved that a triangulation of a
3-ball which contains a knotted spanning arc consisting of one edge
only is not shellable. He also provided a construction to proof the
existence of such a triangulation. The following construction is taken
from Ziegler [3], who uses it to construct an example of a non extendably
shellable 4-polytope. Furthermore Hachimori and Ziegler [2] used
Furch's Ball to obtain nonconstructible d-spheres.

Let *C*_{1} be a pile of 9*7*5
cubes. In the following construction the cube complex
*C*_{1} suffices to obtain a 3-ball which contains a
knotted spanning arc consisting of one edge only. If one wishes a
triangulated 3-ball any triangulation of *C*_{1} will do
and also the provided master file is one. Now *C* is obtained
from *C*_{1} by removing cubes along a knotted spanning
arc (the blue cubes in *figure 1*). If one stops drilling one step before
destroying the property of having a ball, one
arrives at a ball with a knotted spanning edge.

To obtain a non extendably shellable 4-polytope *P*, consider
the following construction: let *V* be the
set of vertices of *C*_{1} and for each vertex
*v* in *V* assign *v*_{1}^{2}
+ v_{2}^{2} + v_{3}^{2} as the fourth
coordinate of *v*. Then define *P := conv(V)* and
*C*_{1} is a subcomplex of the boundary of *P* [3].
*P\C*_{1} is a partial shelling of *P*,
and if one continues this shelling by adding the cubes along the knotted
spanning arc (the blue cubes), one ends up with *P\C* as a partial shelling of
*P*. Every shelling of *P* that shells *P\C* first
would yield a shelling of *C* (Furch's Ball). Since Furch's Ball
is not shellable [1][3], non such shelling exists and *P* is not
extendably shellable. (Note, that in the example described by Ziegler [3]
a pile of 7*5*5 cubes suffices: Since *P* is defined as the convex hull of the vertices of
*C*_{1} (with additional fourth coordinates), facets
are added to *C*_{1} anyway. Therefore some of the cubes
which ensure that the `drilled hole' lies in the interior of
*C*_{1} are not needed to construct the example.)

Hachimori and Ziegler [2] proved that in a constructible 3-ball or 3-sphere, every
knot that consists of three edges (and vertices) is
trivial. (In the following let
*x***F* denote the cone over *F* with apex
*x*.) To obtain a
nonconstructible triangulation of a 3-sphere one
uses *C* (Furch's Ball) and adds the cone over the boundary. The
resulting 3-sphere *C union (v*boundary(C))* has a
non trivial knot that consists of only three edges, therefore it is nonconstructible. If
*C*_{d-1} is a nonconstructible triangulation of a
(*d*-1)-sphere, than *C*_{d} := suspension of
C_{d-1} (*= C*_{d-1} union v*C_{d-1} union
w*C_{d-1} for two vertices *v*, *w* not
contained in *C*_{d-1}) is a nonconstructible
*d*-sphere [2].

The applet (*figure 1*) depicts the cubes to be removed
(colored blue) embedded in the pile of 9*7*5 cubes and a knotted edge
colored yellow.

The master file *Furchs_ball.jvx* contains two
geometries: The 3-ball itself (encoded as the vertex sets of its facets)
and a knotted edge (encoded by its vertices).

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

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

**Zentralblatt No.** | | 05264881 |

#### References

- R. Furch:
*Zur Grundlegung der kombinatorischen Topologie*, Abh. Math. Sem. Hamb. Univ. **3** (1924), 69-88.
- M. Hachimori and G. M. Ziegler:
*Decomposition of simplicial balls and spheres with knots consisting of few edges*, Mathematische Zeitschrift **235** (2000), 159-171.
- G. M. Ziegler: Lectures on Polytopes, Springer (1994).

#### Files

#### Submission information

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

Revised: Thu Apr 11 12:02:59 CEST 2002, Fri Jun 14 18:27:59 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