Equivalence of rational links and 2-bridge links revisited

TL;DR

This paper provides a simple, elementary combinatorial proof demonstrating the equivalence between rational links defined by continued fractions and 2-bridge links, simplifying previous complex proofs involving lens spaces.

Contribution

It introduces an elementary, combinatorial method to establish the equivalence, avoiding the complex topological arguments used in prior proofs.

Findings

Elementary proof of link equivalence

Simplifies understanding of rational and 2-bridge links

Avoids complex topological tools

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 $p/q$.