  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 link. 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.
