  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 $S^3$. 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 $S^3$ after Dehn surgery. Also, using a result of Birman and
Menasco (1994), we determine which closed pure $3$-braids are hyperbolic.
