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

TL;DR

This paper introduces a generalized Schubert normal form for 3-bridge links using six positive integers, and provides explicit formulas for the link group presentations based on this form.

Contribution

It extends the classical Schubert normal form to 3-bridge links and derives concrete group presentations from this new representation.

Findings

Defined the Schubert normal form for 3-bridge links

Provided formulas for the 3-bridge link group presentations

Generalized the link group presentation of 2-bridge links

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 $q$.