Alexander polynomials of simple-ribbon knots

Abstract

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.

Figures (10)

• Figure 1: ribbon knots with less than or equal to nine crossings
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Lemma 1.3
• proof
• Proposition 1.4
• proof
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof