1,569 research outputs found
On a triangulation of the 3-ball and the solid torus
AbstractWe show that neither the 3-ball nor the solid torus admits a triangulation in which (i) every vertex is on the boundary, and (ii) every tetrahedron has exactly one triangle on the boundary. Such triangulations are relevant to an unresolved conjecture of Perles
Face pairing graphs and 3-manifold enumeration
The face pairing graph of a 3-manifold triangulation is a 4-valent graph
denoting which tetrahedron faces are identified with which others. We present a
series of properties that must be satisfied by the face pairing graph of a
closed minimal P^2-irreducible triangulation. In addition we present
constraints upon the combinatorial structure of such a triangulation that can
be deduced from its face pairing graph. These results are then applied to the
enumeration of closed minimal P^2-irreducible 3-manifold triangulations,
leading to a significant improvement in the performance of the enumeration
algorithm. Results are offered for both orientable and non-orientable
triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the
final theorem to the non-orientable case; v3: fixed a flaw in the proof of
the conical face lemm
On the tree-width of knot diagrams
We show that a small tree-decomposition of a knot diagram induces a small
sphere-decomposition of the corresponding knot. This, in turn, implies that the
knot admits a small essential planar meridional surface or a small bridge
sphere. We use this to give the first examples of knots where any diagram has
high tree-width. This answers a question of Burton and of Makowsky and
Mari\~no.Comment: 14 pages, 6 figures. V2: Minor updates to expositio
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
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
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