1-genus 1-bridge splittings for knots
A knot \(K\) in the 3-sphere \(\mathbb S^3\) is called a 1-genus 1-bridge knot if a torus \(H\) splits \(\mathbb S^3\) into two solid tori each of which \(K\) intersects in a trivial arc. The class of 1-genus 1-bridge knots (briefly \((1,1)\)-knots) contains all torus knots and all 2-bridge knots. In this paper the author studies 1-genus 1-bridge splittings of 1-genus 1-bridge knots in the 3-sphere or in a lens space. Such splittings for 2-bridge knots have been studied in [\textit{T. Kobayashi} and \textit{O. Saeki}, Pac. J. Math. 195, 101--156 (2000; Zbl 1019.57010)]. In the paper under review it is proved that if \(K\) is a knot in \(\mathbb S^3\) which is neither a 2-bridge knot nor a satellite knot, and if \(K\) has two non-isotopic 1-genus 1-bridge splitting tori \(H_1\) and \(H_2\), then it is possible to move them to intersect each other in two essential circles, and either \(K\) is obtained from a component of a 2-bridge link by twisting along the other component, or one can move \(H_1\) and \(H_2\) further to meet in a torus with two holes such that each circle \(c_i\) bounds a disc in \(H_i-K\) and a twice punctured disc in \(H_j-K\) for \((i,j)=(1,2)\) and \((2,1)\). More general results which give a detailed description of all cases that can occur also for knots in lens spaces are proved.
scientific article; zbMATH DE number 2111463