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

Lorena Armas-Sanabria; Jesús Rodríguez Viorato; E. Fanny Jasso-Hernández

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arXiv · 2306.09636

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

Lorena Armas-Sanabria·Jesús Rodríguez Viorato·E. Fanny Jasso-Hernández

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Abstract

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.

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Lorena Armas-Sanabria, Jesús Rodríguez Viorato, E. Fanny Jasso-Hernández. 2023-06-16. Artin Presentations of the Trivial Group and Hyperbolic Closed Pure $3$-Braids. https://arxiv.org/abs/2306.09636

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