The Schubert normal form of a 3-bridge link and the 3-bridge link group

Abstract

We introduce the Schubert form a $3$-bridge link diagram, as a generalization of the Schubert normal form of a $3$-bridge link. It consists of a set of six positive integers, written as $\left( p/n,q/m,s/l\right) $, with some conditions and it is based on the concept of $3$-butterfly. Using the Schubert normal form of a $3$-bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers $\left\{ p,n,q,m,s,l\right\} .$ The construction is a generalization of the form the link group presentation of the $2$-bridge link $p/q$ depends on the integers $p$ and $q$.