Furch's Ball

Author

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 C₁ be a pile of 9*7*5 cubes. In the following construction the cube complex C₁ 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 C₁ will do and also the provided master file is one. Now C is obtained from C₁ 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 C₁ and for each vertex v in V assign v₁² + v₂² + v₃² as the fourth coordinate of v. Then define P := conv(V) and C₁ is a subcomplex of the boundary of P [3]. P\C₁ 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 C₁ (with additional fourth coordinates), facets are added to C₁ anyway. Therefore some of the cubes which ensure that the `drilled hole' lies in the interior of C₁ 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).

References

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

Files

• Master File: Furchs_ball.jvx
• Applet File: Furchs_ball_Applet.jvx
• Preview: Furchs_ball_Preview.gif

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

TU Berlin
Fakultät 2, Institut für Mathematik, MA 6-2
Strasse des 17. Juni 136
10623 Berlin, Germany
witte@math.tu-berlin.de