56 research outputs found
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
The number of Reidemeister Moves Needed for Unknotting
There is a positive constant such that for any diagram representing
the unknot, there is a sequence of at most Reidemeister moves that
will convert it to a trivial knot diagram, is the number of crossings in
. A similar result holds for elementary moves on a polygonal knot
embedded in the 1-skeleton of the interior of a compact, orientable,
triangulated 3-manifold . There is a positive constant such that
for each , if consists of tetrahedra, and is unknotted,
then there is a sequence of at most elementary moves in which
transforms to a triangle contained inside one tetrahedron of . We obtain
explicit values for and .Comment: 48 pages, 14 figure
Computational topology with Regina: Algorithms, heuristics and implementations
Regina is a software package for studying 3-manifold triangulations and
normal surfaces. It includes a graphical user interface and Python bindings,
and also supports angle structures, census enumeration, combinatorial
recognition of triangulations, and high-level functions such as 3-sphere
recognition, unknot recognition and connected sum decomposition.
This paper brings 3-manifold topologists up-to-date with Regina as it appears
today, and documents for the first time in the literature some of the key
algorithms, heuristics and implementations that are central to Regina's
performance. These include the all-important simplification heuristics, key
choices of data structures and algorithms to alleviate bottlenecks in normal
surface enumeration, modern implementations of 3-sphere recognition and
connected sum decomposition, and more. We also give some historical background
for the project, including the key role played by Rubinstein in its genesis 15
years ago, and discuss current directions for future development.Comment: 29 pages, 10 figures; v2: minor revisions. To appear in "Geometry &
Topology Down Under", Contemporary Mathematics, AM
Complexity of Seifert manifolds
In this thesis, we give an overview over the theory of Seifert fibre spaces and the complexity theory. We start by giving some preliminary notions about 2-dimensional orbifolds, fibre bundles and circle bundles, in order to be able to understand the following part of the thesis, regarding the theory of Seifert fibre spaces. We first see the definition and properties of Seifert fibre spaces and, after giving a combinatorial description, we classify them up to fibre-preserving homeomorphism and up to homeomorphism. Afterwards, we introduce the complexity theory, at first in a general way concerning all compact 3-manifolds and then focusing ourselves on the estimation for the complexity of Seifert fibre spaces. We also give some examples of spine constructions for manifolds with boundary having complexity zero
GENERAL FLIPS AND THE CD-INDEX
We generalize bistellar operations (often called flips) on simplicial manifolds to a notion of general flips on PL-spheres. We provide methods for computing the cd-index of these general flips, which is the change in the cd-index of any sphere to which the flip is applied. We provide formulas and relations among flips in certain classes, paying special attention to the classic case of bistellar flips. We also consider questions of flip-connecticity , that is, we show that any two polytopes in certain classes can be connected via a sequence of flips in an appropriate class
The topological correctness of PL approximations of isomanifolds
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd â Rdân. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently
fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary
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