51 research outputs found
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Enumerating all 3-manifold triangulations of a given size is a difficult but
increasingly important problem in computational topology. A key difficulty for
enumeration algorithms is that most combinatorial triangulations must be
discarded because they do not represent topological 3-manifolds. In this paper
we show how to preempt bad triangulations by detecting genus in
partially-constructed vertex links, allowing us to prune the enumeration tree
substantially.
The key idea is to manipulate the boundary edges surrounding partial vertex
links using expected logarithmic time operations. Practical testing shows the
resulting enumeration algorithm to be significantly faster, with up to 249x
speed-ups even for small problems where comparisons are feasible. We also
discuss parallelisation, and describe new data sets that have been obtained
using high-performance computing facilities.Comment: 16 pages, 7 figures, 3 tables; v2: minor revisions; to appear in
ISSAC 201
Fixed parameter tractable algorithms in combinatorial topology
To enumerate 3-manifold triangulations with a given property, one typically
begins with a set of potential face pairing graphs (also known as dual
1-skeletons), and then attempts to flesh each graph out into full
triangulations using an exponential-time enumeration. However, asymptotically
most graphs do not result in any 3-manifold triangulation, which leads to
significant "wasted time" in topological enumeration algorithms. Here we give a
new algorithm to determine whether a given face pairing graph supports any
3-manifold triangulation, and show this to be fixed parameter tractable in the
treewidth of the graph.
We extend this result to a "meta-theorem" by defining a broad class of
properties of triangulations, each with a corresponding fixed parameter
tractable existence algorithm. We explicitly implement this algorithm in the
most generic setting, and we identify heuristics that in practice are seen to
mitigate the large constants that so often occur in parameterised complexity,
highlighting the practicality of our techniques.Comment: 16 pages, 9 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
An edge-based framework for enumerating 3-manifold triangulations
A typical census of 3-manifolds contains all manifolds (under various
constraints) that can be triangulated with at most n tetrahedra. Al- though
censuses are useful resources for mathematicians, constructing them is
difficult: the best algorithms to date have not gone beyond n = 12. The
underlying algorithms essentially (i) enumerate all relevant 4-regular
multigraphs on n nodes, and then (ii) for each multigraph G they enumerate
possible 3-manifold triangulations with G as their dual 1-skeleton, of which
there could be exponentially many. In practice, a small number of multigraphs
often dominate the running times of census algorithms: for example, in a
typical census on 10 tetrahedra, almost half of the running time is spent on
just 0.3% of the graphs.
Here we present a new algorithm for stage (ii), which is the computational
bottleneck in this process. The key idea is to build triangulations by
recursively constructing neighbourhoods of edges, in contrast to traditional
algorithms which recursively glue together pairs of tetrahedron faces. We
implement this algorithm, and find experimentally that whilst the overall
performance is mixed, the new algorithm runs significantly faster on those
"pathological" multigraphs for which existing methods are extremely slow. In
this way the old and new algorithms complement one another, and together can
yield significant performance improvements over either method alone.Comment: 29 pages, 19 figure
Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.
In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds.
The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants
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
Recommended from our members
Triangulations
The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods
Finding large counterexamples by selectively exploring the Pachner graph
We often rely on censuses of triangulations to guide our intuition in
-manifold topology. However, this can lead to misplaced faith in conjectures
if the smallest counterexamples are too large to appear in our census. Since
the number of triangulations increases super-exponentially with size, there is
no way to expand a census beyond relatively small triangulations; the current
census only goes up to tetrahedra. Here, we show that it is feasible to
search for large and hard-to-find counterexamples by using heuristics to
selectively (rather than exhaustively) enumerate triangulations. We use this
idea to find counterexamples to three conjectures which ask, for certain
-manifolds, whether one-vertex triangulations always have a "distinctive"
edge that would allow us to recognise the -manifold.Comment: 35 pages, 28 figures. A short version has been accepted for SoCG
2023; this full version contains some new results that do not appear in the
SoCG versio
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