Inflations of ideal triangulations
As summarized in the introduction of the paper persuasively, triangulations play a central role in the study and understanding of \(3\)-manifolds. In [\textit{W. Jaco} and \textit{J. H. Rubinstein}, J. Differ. Geom. 65, No. 1, 61--168 (2003; Zbl 1068.57023)], the authors introduced a technique for crushing a triangulation of a \(3\)-manifold along a normal surface. Later, this was extended to manifolds with boundary, crushing a triangulation along the boundary and arriving at a related ideal triangulation of the interior of the \(3\)-manifold. In this paper, a dual operation for ideal triangulations, called an inflation, is introduced. More precisely, starting from an ideal triangulation of the interior of a compact \(3\)-manifold with boundary, an inflation yields a strongly related triangulation of the compact \(3\)-manifold itself. And crushing along its boundary gives back the original ideal triangulation.NEWLINENEWLINEIn the last part of the paper, two concrete examples of inflations are exhibited. One is that of the two-tetrahedron ideal triangulation of the figure-eight knot complement, and the other one is that of the one-tetrahedron ideal triangulation of the Gieseking manifold, which is a non-orientable \(3\)-manifold doubly covered by the ideal triangulation of the figure-eight knot complement. Applications of inflations will be given in other papers by the same authors.
scientific article; zbMATH DE number 6358435
scientific article