A New Approach to Crushing 3-Manifold Triangulations

Abstract

The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testing for essential surfaces. Although the crushing operation will always reduce the size of a triangulation, it might alter its topology, and so it requires a careful theoretical analysis for the settings in which it is used. The aim of this short paper is to make the crushing operation more accessible to practitioners and easier to generalise to new settings. When the crushing operation was first introduced, the analysis was powerful but extremely complex. Here we give a new treatment that reduces the crushing process to a sequential combination of three "atomic" operations on a cell decomposition, all of which are simple to analyse. As an application, we generalise the crushing operation to the setting of non-orientable 3-manifolds, where we obtain a new practical and robust algorithm for non-orientable prime decomposition. We also apply our crushing techniques to the study of nonorientable minimal triangulations.

On the pathwidth of hyperbolic 3-manifolds

“…In the study of triangulations, their dual graphs play an instrumental role. 9 Given a triangulation T = ∆/Φ, its dual graph Γ(T ) = (V, E) is a multigraph 10 where the nodes in V correspond to the tetrahedra in ∆, and for each face gluing ϕ ∈ Φ identifying two triangular faces of ∆ i and ∆ j , we add an arc between the corresponding nodes in V , cf. Figure 1(ii).…”

“…Maria and Purcell proceed by triangulating the thick part of D using the work of Jørgensen-Thurston [60, §5.11] and Kobayashi-Rieck [33]. This partial triangulation is then simplified [9,28] and completed into the desired triangulation of M.…”

“…More precisely, we only consider topological spaces which are second countable and Hausdorff 9. Following a convention adopted by several authors in the field of computational low-dimensional topology, throughout this paper we use the terms edge and vertex to refer to an edge or vertex in a 3-manifold triangulation, whereas the terms arc and node denote an edge or vertex in a graph, respectively 10.…”