33 research outputs found
Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams
The idea of computing Matveev complexity by using Heegaard decompositions has
been recently developed by two different approaches: the first one for closed
3-manifolds via crystallization theory, yielding the notion of Gem-Matveev
complexity; the other one for compact orientable 3-manifolds via generalized
Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this
paper we extend to the non-orientable case the definition of modified Heegaard
complexity and prove that for closed 3-manifolds Gem-Matveev complexity and
modified Heegaard complexity coincide. Hence, they turn out to be useful
different tools to compute the same upper bound for Matveev complexity.Comment: 12 pages; accepted for publication in Topology and Its Applications,
volume containing Proceedings of Prague Toposym 201
Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra
We present the census of all non-orientable, closed, connected 3-manifolds
admitting a rigid crystallization with at most 30 vertices. In order to obtain
the above result, we generate, manipulate and compare, by suitable computer
procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 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
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
Non-orientable 3-manifolds of small complexity
We classify all closed non-orientable P2-irreducible 3-manifolds having
complexity up to 6 and we describe some having complexity 7. We show in
particular that there is no such manifold with complexity less than 6, and that
those having complexity 6 are precisely the 4 flat non-orientable ones and the
filling of the Gieseking manifold, which is of type Sol. The manifolds having
complexity 7 we describe are Seifert manifolds of type H2 x S1 and a manifold
of type Sol.Comment: 27 pages, 12 figures. Two mistakes contained in the previous version
are fixed: there is a Sol manifold with complexity 6, and the examples with
complexty 7 are Sol and H2xS1 (see the abstract