Bounds on Pachner moves and systoles of cusped 3-manifolds

Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ 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 $M$, we also give a lower bound on the systole length of $M$ 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 $\Delta \times I$ where $\Delta$ is a 2-simplex and $I$ is a closed interval. As an analogue of simplicial complexes, we introduce prism complexes and show that every compact $3$-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 $3$-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 $3$-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 $M$ be a compact 3-manifold with a triangulation $\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. 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 $F$, that depends on the maximum number of tetrahedrons that share a vertex in $\tau$.Comment: 7 pages, 1 figur