Thin position and the recognition problem for \(S^ 3\)
In 1992 Rubinstein gave a series of lectures describing an algorithm to determine whether or not a triangulated 3-manifold is the 3-sphere. His proof uses the language of PL minimal surface theory. In the present paper a different proof is given that a simplified algorithm works using techniques from knot theory. Roughly, the algorithm is as follows. A normal surface in a triangulated 3-manifold \(M\) is an embedded surface intersecting each 3-simplex in a certain simple pattern (nice triangles and quadrilaterals). Now a maximal collection of disjoint non-parallel normal 2-spheres (which can be constructed by a modification of an algorithm due to Haken) cuts \(M\) into three possible types of components two of which are balls resp. punctured balls. Now \(M\) is the 3-sphere if and only if also each component of the third type is a 3-ball which, by the main Lemma, is the case if and only if it contains an ``almost normal'' 2-sphere.
scientific article; zbMATH DE number 817729