RIBBON KNOTS WITH TWO RIBBON TYPES

A ribbon n-knot K n is constructed by attaching m bands to m + 1n-spheres in the euclidean (n + 2)-space. There are many way of attaching them; as a result, K n has many presentations which are called ribbon presentations. But concerning the case of m = 1, it was proved in the case of n = 1 by M. Scharlemann, and n ≥ 2 by Y. Marumoto that if K n is unknotted, its ribbon presentation is essentially unique. In this note, we will prove in the case of m = 1 and n ≥ 2 that there are infinitely many ribbon n-knots which has essentially different two ribbon presentations.