## A shellable 3-ball and 3-sphere with a knot consisting of 4 edges

Electronic Geometry Model No. 2001.05.001#### 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 3 edges is not constructible. 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 5*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 pink vertex *a* and the red vertex *b*
are marked bigger since they are needed for the following: Let
*C*_{2} := *C*_{1} union (*a**(green
faces)) union (*b**(blue faces)), then *C*_{2} is a
3-ball with a knot *ab-bc-cd-da*. Since we add *a**(green
faces) to *C*_{1}, the edge *da* actually exists and
adding *b**(blue faces) creates the edge *bc*. The edges
*ab* and *cd* are already contained in
*C*_{1}. To visualize the knot, one may imagine it as
following the yellow line. This knot *ab-bc-cd-da* is not trivial
because *ab-bc-cd* is a non-trivial knotted spanning arc. It is a
shellable 3-ball because C_{1}
is shellable. Note that we do not form the cone over the dark
pink face adjacent to *a*. Similarly we do not
form the cone over dark red face adjacent to *b*.

In order to get a 3-sphere with a knot consisting of 4
edges, one takes the cone over the boundary of
*C*_{2}, that is *C* := *C*_{2} union
(*v**boundary(*C*_{2})), for *v* 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 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 *shel_3-ball_with_knot.jvx* contains two
geometries: The facets of the 3-ball (encoded as their vertex sets) in
shelling order and the knot (encoded as the vertex sets of its
edges). In the master file point *a* corresponds to 35, *b*
to 34, *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.** | | 05264877 |

#### 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 18:21:37 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