A classification of rational 3-tangles

In this paper, we define the $\mathit{\text{normal}}\mathit{\text{form}}$ and $\mathit{\text{normal}}\mathit{\text{coordinate}}$ of a rational 3-tangle $T$ with respect to $\mathrm{\partial }{E}_1$, where $E_1$ is the fixed two punctured disk in ${\mathrm{\Sigma }}_{0,6}$. Among all normal coordinates of $T$ with respect to $\mathrm{\partial }{E}_1$, we investigate the collection of $\mathit{\text{minimal}}$ 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.