634 research outputs found
A duplicate pair in the SnapPea census
We identify a duplicate pair in the well-known Callahan-Hildebrand-Weeks
census of cusped finite-volume hyperbolic 3-manifolds. Specifically, the
six-tetrahedron non-orientable manifolds x101 and x103 are homeomorphic.Comment: 5 pages, 3 figures; v2: minor edits. To appear in Experimental
Mathematic
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
The complexity of the normal surface solution space
Normal surface theory is a central tool in algorithmic three-dimensional
topology, and the enumeration of vertex normal surfaces is the computational
bottleneck in many important algorithms. However, it is not well understood how
the number of such surfaces grows in relation to the size of the underlying
triangulation. Here we address this problem in both theory and practice. In
theory, we tighten the exponential upper bound substantially; furthermore, we
construct pathological triangulations that prove an exponential bound to be
unavoidable. In practice, we undertake a comprehensive analysis of millions of
triangulations and find that in general the number of vertex normal surfaces is
remarkably small, with strong evidence that our pathological triangulations may
in fact be the worst case scenarios. This analysis is the first of its kind,
and the striking behaviour that we observe has important implications for the
feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2
tables; v2: added minor clarification
Comment on `Lost in Translation: Topological Singularities in Group Field Theory'
Gurau argued in [arXiv:1006.0714] that the gluing spaces arising as Feynman
diagrams of three-dimensional group field theory are not all pseudo-manifolds.
I dispute this conclusion: albeit not properly triangulated, these spaces are
genuine pseudo-manifolds, viz. their singular locus is of codimension at least
two
The complexity of detecting taut angle structures on triangulations
There are many fundamental algorithmic problems on triangulated 3-manifolds
whose complexities are unknown. Here we study the problem of finding a taut
angle structure on a 3-manifold triangulation, whose existence has implications
for both the geometry and combinatorics of the triangulation. We prove that
detecting taut angle structures is NP-complete, but also fixed-parameter
tractable in the treewidth of the face pairing graph of the triangulation.
These results have deeper implications: the core techniques can serve as a
launching point for approaching decision problems such as unknot recognition
and prime decomposition of 3-manifolds.Comment: 22 pages, 10 figures, 3 tables; v2: minor updates. To appear in SODA
2013: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete
Algorithm
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
Optimizing the double description method for normal surface enumeration
Many key algorithms in 3-manifold topology involve the enumeration of normal
surfaces, which is based upon the double description method for finding the
vertices of a convex polytope. Typically we are only interested in a small
subset of these vertices, thus opening the way for substantial optimization.
Here we give an account of the vertex enumeration problem as it applies to
normal surfaces, and present new optimizations that yield strong improvements
in both running time and memory consumption. The resulting algorithms are
tested using the freely available software package Regina.Comment: 27 pages, 12 figures; v2: Removed the 3^n bound from Section 3.3,
fixed the projective equation in Lemma 4.4, clarified "most triangulations"
in the introduction to section 5; v3: replace -ise with -ize for Mathematics
of Computation (note that this changes the title of the paper
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