Equivalence of rational links and 2-bridge links revisited

Abstract: In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction $\left[ a_{1},a_{2},\cdots a_{m}\right],$ $a_{i}\in\mathbb{N}$, and the two bridge link of type $p/q,$ where $p/q$ is the rational given by $\left[ a_{1}%,a_{2},\cdots a_{m}\right] $. The known proof of this equivalence relies on the two fold cover of a link and the classification of the lens spaces. Our proof is elementary and combinatorial and follows the naive approach of finding a set of movements to transform the rational link given by $\left[ a_{1},a_{2},\cdots a_{m}\right] $ into the two bridge link of type $p/q$.