13 research outputs found
simpcomp -- A GAP toolbox for simplicial complexes
simpcomp is an extension (a so called package) to GAP, the well known system
for computational discrete algebra. The package enables the user to compute
numerous properties of (abstract) simplicial complexes, provides functions to
construct new complexes from existing ones and an extensive library of
triangulations of manifolds.Comment: 4 page
Simplicial blowups and discrete normal surfaces in simpcomp
simpcomp is an extension to GAP, the well known system for computational
discrete algebra. It allows the user to work with simplicial complexes. In the
latest version, support for simplicial blowups and discrete normal surfaces was
added, both features unique to simpcomp. Furthermore, new functions for
constructing certain infinite series of triangulations have been implemented
and interfaces to other software packages have been improved to previous
versions.Comment: 10 page
Partitioning the triangles of the cross polytope into surfaces
We present a constructive proof that there exists a decomposition of the
2-skeleton of the k-dimensional cross polytope into closed surfaces
of genus , each with a transitive automorphism group given by the
vertex transitive -action on . Furthermore we show
that for each the 2-skeleton of the (k-1)-simplex is a union
of highly symmetric tori and M\"obius strips.Comment: 13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. /
Contributions to Algebra and Geometry, 53(2):473-486, 201
Simple crystallizations of 4-manifolds
Minimal crystallizations of simply connected PL 4-manifolds are very natural
objects. Many of their topological features are reflected in their
combinatorial structure which, in addition, is preserved under the connected
sum operation. We present a minimal crystallization of the standard PL K3
surface. In combination with known results this yields minimal crystallizations
of all simply connected PL 4-manifolds of "standard" type, that is, all
connected sums of , , and the K3 surface. In
particular, we obtain minimal crystallizations of a pair of homeomorphic but
non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that
the minimal 8-vertex crystallization of is unique and its
associated pseudotriangulation is related to the 9-vertex combinatorial
triangulation of by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear
in Advances in Geometr
Stacked polytopes and tight triangulations of manifolds
Tightness of a triangulated manifold is a topological condition, roughly
meaning that any simplexwise linear embedding of the triangulation into
euclidean space is "as convex as possible". It can thus be understood as a
generalization of the concept of convexity. In even dimensions,
super-neighborliness is known to be a purely combinatorial condition which
implies the tightness of a triangulation.
Here we present other sufficient and purely combinatorial conditions which
can be applied to the odd-dimensional case as well. One of the conditions is
that all vertex links are stacked spheres, which implies that the triangulation
is in Walkup's class . We show that in any dimension
\emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz
are tight.
Furthermore, triangulations with -stacked vertex links and the centrally
symmetric case are discussed.Comment: 28 pages, 2 figure
Combinatorial properties of the K3 surface: Simplicial blowups and slicings
The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3
surface by resolving the 16 singularities. Here we present a simplicial
realization of this minimal resolution. Starting with a minimal 16-vertex
triangulation of K^4 we resolve its 16 isolated singularities - step by step -
by simplicial blowups. As a result we obtain a 17-vertex triangulation of the
standard PL K3 surface. A key step is the construction of a triangulated
version of the mapping cylinder of the Hopf map from the real projective
3-space onto the 2-sphere with the minimum number of vertices. Moreover we
study simplicial Morse functions and the changes of their levels between the
critical points. In this way we obtain slicings through the K3 surface of
various topological types.Comment: 31 pages, 3 figure
Combinatorial Seifert fibred spaces with transitive cyclic automorphism group
In combinatorial topology we aim to triangulate manifolds such that their
topological properties are reflected in the combinatorial structure of their
description. Here, we give a combinatorial criterion on when exactly
triangulations of 3-manifolds with transitive cyclic symmetry can be
generalised to an infinite family of such triangulations with similarly strong
combinatorial properties. In particular, we construct triangulations of Seifert
fibred spaces with transitive cyclic symmetry where the symmetry preserves the
fibres and acts non-trivially on the homology of the spaces. The triangulations
include the Brieskorn homology spheres , the lens spaces
and, as a limit case, .Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of
Mathematic
Combinatorial 3-manifolds with transitive cyclic symmetry
In this article we give combinatorial criteria to decide whether a transitive
cyclic combinatorial d-manifold can be generalized to an infinite family of
such complexes, together with an explicit construction in the case that such a
family exists. In addition, we substantially extend the classification of
combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices.
Finally, a combination of these results is used to describe new infinite
families of transitive cyclic combinatorial manifolds and in particular a
family of neighborly combinatorial lens spaces of infinitely many distinct
topological types.Comment: 24 pages, 5 figures. Journal-ref: Discrete and Computational
Geometry, 51(2):394-426, 201