On classification of genus g knots which admit a (1,1)-decomposition

Abstract

Given an oriented minimal genus Seifert surface [Formula: see text] for a [Formula: see text]-knot [Formula: see text] it is possible to surger [Formula: see text] along annuli to obtain a simple minimal Seifert surface[Formula: see text] . Such a surface can be put in a very nice position with respect to the [Formula: see text]-position of the knot [Formula: see text]. Using this kind of surfaces we give a description of a [Formula: see text]-knot of genus [Formula: see text] as a vertical banding of [Formula: see text]-knots of genus smaller than [Formula: see text]. In addition, we show that any rational knot of genus [Formula: see text] is obtained as a vertical banding of [Formula: see text] genus one rational knots.