160 research outputs found
Computing Invariants of Simplicial Manifolds
This is a survey of known algorithms in algebraic topology with a focus on
finite simplicial complexes and, in particular, simplicial manifolds. Wherever
possible an elementary approach is chosen. This way the text may also serve as
a condensed but very basic introduction to the algebraic topology of simplicial
manifolds.
This text will appear as a chapter in the forthcoming book "Triangulated
Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
Convergent Puiseux Series and Tropical Geometry of Higher Rank
We propose to study the tropical geometry specifically arising from
convergent Puiseux series in multiple indeterminates. One application is a new
view on stable intersections of tropical hypersurfaces. Another one is the
study of families of ordinary convex polytopes depending on more than one
parameter through tropical geometry. This includes cubes constructed by
Goldfarb and Sit (1979) as special cases.Comment: 32 pages, 3 figure
Branched Coverings, Triangulations, and 3-Manifolds
A canonical branched covering over each sufficiently good simplicial complex
is constructed. Its structure depends on the combinatorial type of the complex.
In this way, each closed orientable 3-manifold arises as a branched covering
over the 3-sphere from some triangulation of S^3. This result is related to a
theorem of Hilden and Montesinos. The branched coverings introduced admit a
rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
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