## A shellable 3-ball with a knotted spanning arc consisting of 3 edges

Electronic Geometry Model No. 2001.05.002#### 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 2 edges is not constructible. The following
example (constructed along the ideas of Hachimori and Ziegler [1], yet
considerably smaller) shows that 2 is the best possible upper bound
for the number of edges.

Let *C*_{1} be the following
triangulation of a pile of 4*4*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*. The red vertex *b* is marked bigger since it is needed
for the following: Let *C* := *C*_{1} union
(*b**(blue faces)), then *C* is a shellable 3-ball because
*C*_{1} is shellable, and the arc *ab-bc-cd* is a
knotted spanning arc of *C*. Since we add *b**(blue faces)
to *C*_{1}, the edge *bc* actually exists. 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. Note that we do not form the cone over the dark red face adjacent
to *b*.

*Figure 1* depicts the pile of cubes and the
faces needed for the construction are colored blue. The yellow line
visualizes the knotted arc. To obtain the
example (*shel_3-ball_with_knotted_arc.jvx*) one has to compleat the
construction by adding *b**(blue faces).

The master file *shel_3-ball_with_knotted_arc.jvx* contains
two geometries: The facets of the 3-ball (encoded as their vertex sets) in
shelling order and the knotted spanning arc (encoded as the vertex sets of its
edges). In the master file point *a* corresponds to 25, *b*
to 24, *c* to 15 and *d* to 14. (If needed,
check the applet file for the indices of the other vertices.)

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

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

**Zentralblatt No.** | | 05264878 |

#### 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 18:18:18 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