A 3-sphere with a knotted triangle

Lickorish (1991) showed the there are non-shellable triangulations of a 3-sphere. What he showed is the fact that if a triangulation of a 3-sphere contain a knot made of 3 edges in the 1-skeleton, then it is not shellable. What he suggested to make such a triangulation is to use the following construction. First, we prepare the Furch's 3-ball with a knotted spanning arc consisting of one edge. Then we take a cone over the boundary of the ball, then we have a triangulated 3-sphere with a knot made of 3 edges.

The data given here is made from knot.dat by one-point compactification. This has 381vertices and 1928 facets.

Properties

Lickorish's original theorem asserts that the triangulation is non-shellable if the knot embedded is "complex enough", but in fact, it is not shellable (even not constructible) if the knot is nontrivial.

Datum

nc_sphere.dat

Table

vertex decomposable?	no
extendably shellable?	no
shellable?	no
constructible?	no
Cohen-Macaulay?	yes
topology	3-sphere
f-vector	(1,381,2309,3856,1928)
h-vector	(1,377,1172,377,1)
made by	Lickorish (またはfolklore)

References

W.B.R.Lickorish, Unshellable triangulations of spheres, Europ. J. Cominatorics 12 (1991), 527-530.

M.Hachimori and G.M.Ziegler, Decompositions of balls and spheres with knots consisting of few edges, Math. Z., to appear.

Remark:
A sophisticated triangulation using this knot idea is provided by F. Lutz in:

• F.H. Lutz, Small examples of non-constructible simplicial balls and spheres, SIAM J. Discrete Math. 18, 103-109 (2004).