205 research outputs found
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
Quadrilateral-octagon coordinates for almost normal surfaces
Normal and almost normal surfaces are essential tools for algorithmic
3-manifold topology, but to use them requires exponentially slow enumeration
algorithms in a high-dimensional vector space. The quadrilateral coordinates of
Tollefson alleviate this problem considerably for normal surfaces, by reducing
the dimension of this vector space from 7n to 3n (where n is the complexity of
the underlying triangulation). Here we develop an analogous theory for
octagonal almost normal surfaces, using quadrilateral and octagon coordinates
to reduce this dimension from 10n to 6n. As an application, we show that
quadrilateral-octagon coordinates can be used exclusively in the streamlined
3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing
experimental running times by factors of thousands. We also introduce joint
coordinates, a system with only 3n dimensions for octagonal almost normal
surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using
cohomology, plus other minor changes; v3: Minor housekeepin
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
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
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
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