Furch  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 , who uses it to construct an example of a non extendably shellable 4-polytope. Furthermore Hachimori and Ziegler  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 . 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 , non such shelling exists and P is not extendably shellable. (Note, that in the example described by Ziegler  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  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 Cd-1 is a nonconstructible triangulation of a (d-1)-sphere, than Cd := suspension of Cd-1 (= Cd-1 union v*Cd-1 union w*Cd-1 for two vertices v, w not contained in Cd-1) is a nonconstructible d-sphere .
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)|
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.
Fakultät 2, Institut für Mathematik, MA 6-2
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