Alexander polynomials of simple-ribbon knots

Kengo Kishimoto, Tetsuo Shibuya, Tatsuya Tsukamoto, Tsuneo Ishikawa

Published 2019-05-13Version 1

In a previous paper, 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 <10 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 a p-simple-ribbon knot if the knot is obtained from the trivial knot by a finite sequence of elementary p-simple-ribbon fusions for a fixed positive integer p. 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.