94 research outputs found
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
Computing trisections of 4-manifolds
Algorithms that decompose a manifold into simple pieces reveal the geometric
and topological structure of the manifold, showing how complicated structures
are constructed from simple building blocks. This note describes a way to
algorithmically construct a trisection, which describes a -dimensional
manifold as a union of three -dimensional handlebodies. The complexity of
the -manifold is captured in a collection of curves on a surface, which
guide the gluing of the handelbodies. The algorithm begins with a description
of a manifold as a union of pentachora, or -dimensional simplices. It
transforms this description into a trisection. This results in the first
explicit complexity bounds for the trisection genus of a -manifold in terms
of the number of pentachora (-simplices) in a triangulation.Comment: 15 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
The triangulation complexity of elliptic and sol 3-manifolds
The triangulation complexity of a compact 3-manifold M is the minimal number of tetrahedra in any triangulation of M. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a universally bounded multiplicative error
The triangulation complexity of elliptic and sol 3-manifolds
The triangulation complexity of a compact 3-manifold is the minimal number of
tetrahedra in any triangulation of the 3-manifold. We compute the triangulation
complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a
universally bounded multiplicative error.Comment: 41 pages, 9 figure
Two-dimensional state sum models and spin structures
The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai
are generalised by allowing algebraic data from a non-symmetric Frobenius
algebra. Without any further data, this leads to a state sum model on the
sphere. When the data is augmented with a crossing map, the partition function
is defined for any oriented surface with a spin structure. An algebraic
condition that is necessary for the state sum model to be sensitive to spin
structure is determined. Some examples of state sum models that distinguish
topologically-inequivalent spin structures are calculated.Comment: 43 pages. Mathematica script in ancillary file. v2: nomenclature of
models and their properties changed, some proofs simplified, more detailed
explanations. v3: extended introduction, presentational improvements; final
versio
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