# A Tabulation of Ribbon Knots in Tangle Form

## TL;DR

This paper characterizes all ribbon knots with up to 10 crossings through specific 4-strand tangles, providing explicit diagrams and demonstrating their properties via two types of closures.

## Contribution

It introduces a method to represent all known small ribbon knots using particular tangles with defined closure properties, expanding understanding of their structure.

## Key findings

- Explicit 4-strand tangles for 21 ribbon knots are provided.
- Each tangle satisfies two closure properties related to the knot and unlink.
- Diagrams and planar notations of tangles are included.

## Abstract

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.