Furch's knotted hole ball

Description

A famous example of non-shellable triangulation of a 3-ball. This is a triangulation which has a knotted spanning arc consisting of one edge in its 1-skeleton. The construction is described in Furch's 1924 article.
A "knotted spanning arc" is an arc such as described in the right figure below. The left figure indicates how to embed such a knotted arc by one edge. First we prepare a big pile of small cubes and dig a hole from the bottom face, making a knot as in the figure. If we stop digging one step before the tunnel go through the upper face, we have the bold edge to be what we want. (And just triangulate each cube into six tetrahedra withoug introducing new vertices.)

The data given here is basically according to what described above. (Removed some tetrahedra which are not needed.) It uses 380 vertices and 1172 facets.

Properties

Originally this is known as an example of a non-shellable 3-ball, but in fact, stronger property can be shown that this is not constructible.

Datum

knot.dat

Some table

vertex decomposable?	no
extendably shellable?	no
shellable?	no
constructible?	no
Cohen-Macaulay?	yes
topology	3-ball
f-vector	(1,380,1929,2722,1172)
h-vector	(1,376,795,0,0)
made by	Furch

References

G.M.Ziegler, Shelling polyhedral 3-balls and 4-polytopes, Discrete Comput. Geom. 19 (1998), 159-174.

M.Hachimori, Nonconstructible simplicial balls and a way of testing constructibility, Discrete Comput. Geom. 22 (1999), 223-230.

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