2,298 research outputs found
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
Computational knot theory and 3-manifold topology have seen significant
breakthroughs in recent years, despite the fact that many key algorithms have
complexity bounds that are exponential or greater. In this setting,
experimentation is essential for understanding the limits of practicality, as
well as for gauging the relative merits of competing algorithms.
In this paper we focus on normal surface theory, a key tool that appears
throughout low-dimensional topology. Stepping beyond the well-studied problem
of computing vertex normal surfaces (essentially extreme rays of a polyhedral
cone), we turn our attention to the more complex task of computing fundamental
normal surfaces (essentially an integral basis for such a cone). We develop,
implement and experimentally compare a primal and a dual algorithm, both of
which combine domain-specific techniques with classical Hilbert basis
algorithms. Our experiments indicate that we can solve extremely large problems
that were once though intractable. As a practical application of our
techniques, we fill gaps from the KnotInfo database by computing 398
previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict
attention to primal & dual algorithms only, larger and more detailed
experiments, more new crosscap number
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
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we
improve the census algorithm for enumerating closed minimal P^2-irreducible
3-manifold triangulations. In particular, new constraints are proven for face
pairing graphs, and pruning techniques are improved using a modification of the
union-find algorithm. Using these results we catalogue all 136 closed
non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten
tetrahedra.Comment: 37 pages, 34 figure
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
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
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
Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census
This is a collection of notes on embedding problems for 3-manifolds. The main
question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The
terrain of exploration is the Burton/Martelli/Matveev/Petronio census of
triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There
are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400
orientable manifolds, only 149 of them have hyperbolic torsion linking forms
and are thus candidates for embedability in the 4-sphere. The majority of this
paper is devoted to the embedding problem for these 149 manifolds. At present
41 are known to embed. Among the remaining manifolds, embeddings into homotopy
4-spheres are constructed for 4. 67 manifolds are known to not embed in the
4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric
manifolds i.e. having a trivial JSJ-decomposition.Comment: 58 pages, 80+ figures. V6: Included references to libraries valid in
Regina 5.0+. Incorporated changes suggested by Ahmed Issa, following from his
techniques developed with McCoy. Included a few recent references. To appear
in Experimental Mathematic
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