In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction $left[ a_{1},a_{2},cdots a_{m}right],$ $a_{i}inmathbb{N}$, 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 $p/q$.
With the idea of an eventual classification of 3-bridge links, we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of edges is a li
nk. Several examples are given. We prove that every link can be represented in this way (butterfly representation). We define the butterfly number of a link, and we show that the butterfly number and the bridge number of a link coincide. This is done by defining a move on the butterfly diagram. We give an example of two different butterflies with minimal butterfly number representing the knot $8_{20}.$ This raises the problem of finding a set of moves on a butterfly diagram connecting diagrams representing the same link. This is left as an open problem.
