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 [a1, a2,…,am], ai ∈ ℕ, and the 2-bridge link of type p/q, where p/q is the rational number given by [a1, a2,…,am]. The known proof of this equivalence relies on the 2-fold cover of a link and the classification of the lens spaces. Our proof is elementary and combinatorial and follows the naïve approach of finding a set of movements to transform the rational link given by [a1, a2,…,am] into the 2-bridge link of type p/q.