A classification of rational 3-tangles

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.