ALEXANDER POLYNOMIALS OF SIMPLE-RIBBON KNOTS

In [4], we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <= 9 crossings is a simple-ribbon knot. In this paper, we give a formula for the Alexander polynomials of simple-ribbon knots. Using the formula, we determine if a knot with 10 crossings is a simple-ribbon knot. Every simple-ribbon fusion can be realized by "elementary" simple-ribbon fusions. We call a knot an m-simple-ribbon knot if the knot is obtained from the trivial knot by a finite sequence of elementary m-simple-ribbon fusions for a fixed positive integer m. We provide a condition for a simple-ribbon knot to be both of an m-simple-ribbon knot and an n-simple-ribbon knot for positive integers m and n.