Genus one knots which admit (1,1)-decompositions (Q2782649)

Let \(M\) be either \(S^3\) or a lens space. A knot \(K\subset{}M\) is said to admit a \((1,1)\)-decomposition (or be a \((1,1)\)-knot, for short) if there exists a torus \(H\subset{}M\) splitting \(M\) into two solid tori \(V_1\) and \(V_2\), for which \(K\) intersects \(H\) transversely in two points and \(K\cap{}V_i\) is a trivial arc in \(V_i\) (\(i=1,2\)). NEWLINENEWLINENEWLINEA knot \(K\subset{}S^3\) is said to have tunnel number one if there exists an arc \(\tau\subset{}S^3\) for which \(\tau\cap{}K=\partial\tau\) while removing a tubular neighborhood of \(\tau\cup{}K\) from \(S^3\) leaves a genus two handlebody. NEWLINENEWLINENEWLINEAll \((1,1)\)-knots in \(S^3\) are tunnel number one though the converse does not hold [\textit{K. Morimoto, M. Sakuma}, and \textit{Y. Yokota}, Math. Proc. Camb. Philos. Soc. 119, No. 1, 113-118 (1996; Zbl 0866.57004)]. However satellite tunnel number one knots are all \((1,1)\)-knots, and have been classified completely in the form \(K(\alpha,\beta;p,q)\) by four integer parameters \(\alpha\), \(\beta\), \(p\) and \(q\), by [\textit{K. Morimoto} and \textit{M. Sakuma}, Math. Ann. 289, No. 1, 143-167 (1991; Zbl 0697.57002)]. NEWLINENEWLINENEWLINEThe main theorem of the paper states that any genus one knot in \(S^3\) which admits a \((1,1)\)-decomposition is either a 2-bridge knot of type \(K(4\alpha\beta-1,2\alpha)\) or a satellite tunnel number one knot of type \(K(8m,4m+1;p,q)\). NEWLINENEWLINENEWLINEIn a recent preprint Scharlemann used this theorem to settle a conjecture of \textit{H. Goda} and \textit{M. Teragaito} [Tokyo J. Math. 22, No. 1, 99-103 (1999; Zbl 0939.57009)] showing that non-satellite genus one tunnel number one knots in \(S^3\) are 2-bridge.