8 research outputs found
Small examples of non-constructible simplicial balls and spheres
We construct non-constructible simplicial -spheres with vertices
and non-constructible, non-realizable simplicial -balls with vertices
for .Comment: 9 pages, 3 figure
Knots in collapsible and non-collapsible balls
We construct the first explicit example of a simplicial 3-ball B_{15,66} that
is not collapsible. It has only 15 vertices. We exhibit a second 3-ball
B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable.
Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125}
(with only 18 vertices) that is not locally constructible. All these examples
are based on knotted subcomplexes with only three edges; the knots are the
trefoil, the double trefoil, and the triple trefoil, respectively. The more
complicated the knot is, the more distant the triangulation is from being
polytopal, collapsible, etc. Further consequences of our work are:
(1) Unshellable 3-spheres may have vertex-decomposable barycentric
subdivisions.
(This shows the strictness of an implication proven by Billera and Provan.)
(2) For d-balls, vertex-decomposable implies non-evasive implies collapsible,
and for d=3 all implications are strict.
(This answers a question by Barmak.)
(3) Locally constructible 3-balls may contain a double trefoil knot as a
3-edge subcomplex.
(This improves a result of Benedetti and Ziegler.)
(4) Rudin's ball is non-evasive.Comment: 25 pages, 5 figures, 11 tables, references update
Saturated simplicial complexes
AbstractAmong shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. We extend the notion of saturation to arbitrary pure complexes and give a survey of their properties. It is shown that saturated complexes can be characterized via the p-rank of incidence matrices and via the structure of links. We show that rank-selected subcomplexes of saturated complexes are also saturated, and that order complexes of geometric lattices are saturated
Angle sums on polytopes and polytopal complexes
We will study the angle sums of polytopes, listed in the -vector,
working to exploit the analogy between the f-vector of faces in each dimension
and the alpha-vector of angle sums. The Gram and Perles relations on the
-vector are analogous to the Euler and Dehn-Sommerville relations on
the f-vector. First we describe the spaces spanned by the the alpha-vector and
the -f-vectors of certain classes of polytopes. Families of polytopes
are constructed whose angle sums span the spaces of polytopes defined by the
Gram and Perles equations. This shows that the dimension of the affine span of
the space of angle sums of simplices is floor[(d-1)/2], and that of the
combined angle sums and face numbers of simplicial polytopes and general
polytopes are d-1 and 2d-3, respectively. Next we consider angle sums of
polytopal complexes. We define the angle characteristic on the alpha-vector in
analogy to the Euler characteristic. We show that the changes in the two
correspond and that, in the case of certain odd-dimensional polytopal
complexes, the angle characteristic is half the Euler characteristic. Finally,
we consider spherical and hyperbolic polytopes and polytopal complexes.
Spherical and hyperbolic analogs of the Gram relation and a spherical analog of
the Perles relation are known, and we show the hyperbolic analog of the Perles
relations in a number of cases. Proving this relation for simplices of
dimension greater than 3 would finish the proof of this result. Also, we show
how constructions on spherical and hyperbolic polytopes lead to corresponding
changes in the angle characteristic and Euler characteristic.Comment: Ph.D. Dissertatio