On the classification of a large set of rational 3-tangle diagrams

TL;DR

This paper investigates the classification of a large subset of rational 3-tangle diagrams generated by three elements, addressing the gap in complete classification similar to rational 2-tangles.

Contribution

It introduces a classification for a significant subset of rational 3-tangle diagrams generated by three generators, expanding understanding beyond existing algorithms.

Findings

Identifies a large class of rational 3-tangle diagrams

Provides a classification method for these diagrams

Advances the understanding of tangle diagram distinctions

Abstract

We note that a rational $3$-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational $3$-tangle diagrams up to isotopy. However, there is no perfect classification about rational $3$-tangle diagrams such as the classification of rational $2$-tangle diagrams corresponding to rational numbers. In this paper, we classify a large set of rational $3$-tangles which are generated by only three generators.