A classification of rational 3-tangles

TL;DR

This paper introduces a new classification method for rational 3-tangles using normal forms and coordinates, demonstrating the contractibility of a related simplicial complex to classify tangles up to isotopy.

Contribution

It defines normal forms and coordinates for rational 3-tangles and proves the contractibility of the associated simplicial complex, enabling classification up to isotopy.

Findings

The simplicial complex of normal forms is contractible.

Minimal normal coordinates can be chosen systematically.

The classification of rational 3-tangles up to isotopy is achieved.

Abstract

In this paper, we define the \textit{normal form} and \textit{normal coordinate} of a rational 3-tangle $T$ with respect to $\partial E_1$, where $E_1$ is the fixed two punctured disk in $\Sigma_{0,6}$. Among all normal coordinates of $T$ with respect to $\partial E_1$, we investigate the collection of \textit{minimal} normal coordinates of $T$. We show that the simplicial complex constructed with normal forms of the rational 3-tangle is contractible. As an effectiveness of the contractibility of the simplicial complex by normal forms of $T$, we would choose a minimal normal coordinate of $T$ with a certain rule for the representative for the rational $3$-tangle $T$. This classifies rational $3$-tangles up to isotopy.