Parabolic generating pairs of genus-one 2-bridge knot groups (Q2805388)

A \(2\)-bridge link \(K\) admits a diagram which has a \((\mathbb{Z}_2 )^2\)-symmetry generated by the \(\pi\)-rotations \(h_1\) (resp. \(h_2\)) about the horizontal (resp. vertical) axis in the projection plane such that \(\mathrm{Fix}(h_1 )\) (resp. \(\mathrm{Fix}(h_2 )\)) contains the upper (resp. lower) tunnel. The \textit{long upper (resp. lower) meridian pair} is obtained by joining the arcs of \(\mathrm{Fix}(h_i )\cap\) exterior\(K\) that are different from the upper (resp. lower) tunnel with small meridian loops passing through the endpoints of the arcs. The authors give a combinatorial proof, using small cancellation theory, that for a genus-one \(2\)-bridge knot, the long upper (or long lower) meridian pair generates a free subgroup of the knot group. They use this to obtain an alternative proof of a result of Agol, which is that any parabolic generating pair of a genus-one hyperbolic \(2\)-bridge knot group is equivalent to the upper or lower meridian pair. As an application they obtain a complete classification of the epimorphisms from \(2\)-bridge knot groups to genus-one hyperbolic \(2\)-bridge knot groups.