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
What he suggested to make such a triangulation is to use the
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.
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.
|made by||Lickorish ($B$^$?$O(Bfolklore)|
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.
A sophisticated triangulation using this knot idea is provided by F. Lutz in:
It uses only 13 vertices and 56 facets!
F.H. Lutz, Small examples of non-constructible simplicial balls and spheres,
SIAM J. Discrete Math. 18, 103-109 (2004).