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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 whenthe 3-manifold obtained after performing integral Dehn surgery on closed pure3-braids is $S^3$. In the present paper, we analyze the symmetries of thehexatangle and give a list of Artin $n$-presentations for the trivial group.These presentations correspond to the double-branched covers of the hexatanglethat produce $S^3$ after Dehn surgery. Also, using a result of Birman andMenasco (1994), we determine which closed pure $3$-braids are hyperbolic.

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