On some classes of hyperbolic knots with the 3-sphere surgery

For the distance of ( 1 , 1 ) -splittings of a knot in a closed orientable 3-manifold, it is an important problem whether a ( 1 , 1 ) -knot can admit ( 1 , 1 ) -splittings of different distances. In this paper, we give one-parameter families of hyperbolic ( 1 , 1 ) -knots such that each ( 1 , 1 ) -knot admits a Dehn surgery yielding the 3-sphere. It is remarkable that such knots are the first concrete examples each of whose ( 1 , 1 ) -splittings is of distance three.