Artin presentations of the trivial group and hyperbolic closed pure 3-braids (Q6581829)

The main object of study in the paper under review is a class of links represented by the closure of pure 3-braids, and related to these links, two results are mainly obtained. In the first part of the paper, based on the results on integral Dehn surgeries on such links obtained in [\textit{L. Armas-Sanabria} and \textit{M. Eudave-Muñoz}, Topology Appl. 156, No. 6, 1037--1053 (2009; Zbl 1170.57004)], a list of Artin \(n\)-presentations for the trivial group is given (Tables 5, 6, 7, 8, 9, and 10). Artin \(n\)-presentations of groups were considered by \textit{F. J. González-Acuña} in [3-dimensional open books. Lectures Notes, Univ. of Iowa Topology Seminar (1975)], and those for the trivial group are shown to be closely related to the Poincaré Conjecture. In the second part, the authors determine which closed pure 3-braids are hyperbolic by using the results obtained in [\textit{J. S. Birman} and \textit{W. W. Menasco}, Topology 33, No. 3, 525--556 (1994; Zbl 0833.57004)].