Genus one 1-bridge knots as viewed from the curve complex
In the paper ``3-manifolds as viewed from the curve complex'' [Topology 40, No. 3, 631--657 (2001; Zbl 0985.57014)], \textit{J. Hempel} studied Heegaard splittings of closed 3-manifolds by using the curve complex associated to Heegaard surfaces. In particular, Hempel showed that a Haken manifold is hyperbolic if a Heegaard splitting of the manifold has distance \(\geq 3\). In this paper, a topological characterization of the knots admitting \((1, 1)\)-splittings of distance \(\leq 2\) is given. This characterization implies, as a corollary, that a \((1,1)\)-knot is hyperbolic if and only if has a \((1,1)\)-splitting of distance \(\geq 3\). It is also proved that there are \((1,1)\)-splittings with arbitrarily high distance.
scientific article; zbMATH DE number 2111464