A NEW APPROACH TO CRUSHING 3-MANIFOLD TRIANGULATIONS

Abstract. The crushing operation of Jaco and Rubinstein is a powerful tech-nique in algorithmic 3-manifold topology: it enabled the first practical imple-mentations of 3-sphere recognition and prime decomposition of orientable man-ifolds, 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 accessi-ble to practitioners, and easier to generalise to new settings. When the crush-ing 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 crush-ing 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 non-orientable minimal triangulations. 1