On detecting the trivial rational $3$-tangle

Abstract

An important issue in classifying the rational $3$-tangle is how to know whether or not the given tangle is the trivial rational 3-tangle called $\infty$-tangle. The author\cite{1} provided a certain algorithm to detect the $\infty$-tangle. In this paper, we give a much simpler method to detect the $\infty$-tangle by using the $\text{bridge arc replacement}$. We hope that this method can help prove many application problems such as the classification of $3$-bridge knots.