Alexander polynomials of simple-ribbon knots
In a previous work, the authors introduced special types of fusions, so called simple-ribbon fusions on links. Every simple-ribbon fusion can be realized by ``elementary'' simple-ribbon fusions. 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 \(\leq 9\) crossings is a simple-ribbon knot. An \(m\)-simple-ribbon knot is a knot obtained from the trivial one by a finite sequence of elementary \(m\)-simple-ribbon fusions for a fixed positive integer \(m\). In this paper, the authors give a formula for the Alexander polynomials of simple-ribbon knots. As an application, they determine if a knot with \(10\) crossings is a simple-ribbon knot. Moreover, they 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 \(n\).
scientific article; zbMATH DE number 7383195