# Alexander polynomials of simple-ribbon knots

## TL;DR

This paper derives a formula for Alexander polynomials of simple-ribbon knots, enabling identification of such knots among 10-crossing knots and exploring their fusion properties.

## Contribution

It provides a new formula for Alexander polynomials of simple-ribbon knots and characterizes their fusion structures, advancing understanding of ribbon knot classifications.

## Key findings

- Formula for Alexander polynomials of simple-ribbon knots
- Criterion to identify 10-crossing simple-ribbon knots
- Conditions for knots to be both m- and n-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.