Artin Presentations of the Trivial Group and Hyperbolic Closed Pure 3-Braids


We consider a special class of framed links that arise from the hexatangle. Such links are introduced in arXiv:0807.1677, which was also analyzed when the 3-manifold obtained after performing integral Dehn surgery on closed pure 3-braids is S3. In the present paper, we analyze the symmetries of the hexatangle and give a list of Artin n-presentations for the trivial group. These presentations correspond to the double-branched covers of the hexatangle that produce S3 after Dehn surgery. Also, using a result of Birman and Menasco (1994), we determine which closed pure 3-braids are hyperbolic.