A survey of butterfly diagrams for knots and links

A “butterfly diagram” is a representation of a knot as a kind of graph on the sphere. This generalization of Thurston’s construction of the Borromean rings was introduced by Hilden, Montesinos, Tejada, and Toro to study the bridge number of knots. In this paper, we study various properties of butterfly diagrams for knots and links. We prove basic some combinatorial results about butterflies and explore properties of butterflies for classes of links, especially torus links. The Wirtinger presentation for the knot group will be adapted to butterfly diagrams, and we translate the Reidemeister moves for knot diagrams into so-called “butterfly moves.” The main results of this paper are proofs for the classifications of 1- and 2-bridge links using butterflies. In particular, we prove that a link has bridge number equal to two if and only if it is a rational link. Our proof of this result requires the use of an object which we call a weave. We prove that a weave is equivalent to a rational tangle, and vice-versa. We conclude with a brief discussion of some open questions involving butterfly diagrams