A new representation of Links: Butterflies

TL;DR

This paper introduces butterfly representations for links, proving every link can be represented this way and establishing the equivalence of butterfly number and bridge number, with implications for link classification.

Contribution

It defines butterfly representations for links, proves their universality, and shows the butterfly number equals the bridge number, introducing a new approach to link classification.

Findings

Every link can be represented as a butterfly.

Butterfly number equals bridge number.

Examples of minimal butterfly representations are provided.

Abstract

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.