11 research outputs found
Bounds on Pachner moves and systoles of cusped 3-manifolds
Any two geometric ideal triangulations of a cusped complete hyperbolic
-manifold are related by a sequence of Pachner moves through topological
triangulations. We give a bound on the length of this sequence in terms of the
total number of tetrahedra and a lower bound on dihedral angles. This leads to
a naive but effective algorithm to check if two hyperbolic knots are
equivalent, given geometric ideal triangulations of their complements. Given a
geometric ideal triangulation of , we also give a lower bound on the systole
length of in terms of the number of tetrahedra and a lower bound on
dihedral angles.Comment: Exactly the same arguments work for hyperbolic manifolds with
multiple cusps, so statements of theorems are generalised from one-cusped
hyperbolic manifolds to cusped hyperbolic manifold
Prism complexes
A prism is the product space where is a 2-simplex
and is a closed interval. As an analogue of simplicial complexes, we
introduce prism complexes and show that every compact -manifold has a prism
complex structure. We call a prism complex special if each interior horizontal
edge lies in four prisms, each boundary horizontal edge lies in two prisms and
no horizontal face lies on the boundary. We give a criteria for existence of
horizontal surfaces in (possibly non-orientable) Seifert fibered spaces. Using
this we show that a compact -manifold admits a special prism complex
structure if and only if it is a Seifert fibered space with non-empty boundary,
a Seifert fibered space with a non-empty collection of surfaces in its
exceptional set or a closed Seifert fibered space with Euler number zero. So in
particular, a compact -manifold with boundary is a Seifert fibered space if
and only if it has a special prism complex structure.Comment: Exposition improved. Minor errors correcte
Euler characteristic and quadrilaterals of normal surfaces
Let be a compact 3-manifold with a triangulation . We give an
inequality relating the Euler characteristic of a surface normally embedded
in with the number of normal quadrilaterals in . This gives a relation
between a topological invariant of the surface and a quantity derived from its
combinatorial description. Secondly, we obtain an inequality relating the
number of normal triangles and normal quadrilaterals of , that depends on
the maximum number of tetrahedrons that share a vertex in .Comment: 7 pages, 1 figur