On knots that divide ribbon knotted surfaces

Abstract We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K , the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the double slice genus. We also introduce the half fusion number of a knot K , that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded below by the Levine–Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.