Minimum crossing numbers for 3-braids

Abstract

Given a braid on N strings, find an algorithm which generates an Artin braid word B of minimal length. This is an important unsolved problem-a solution would give us the most economical way of notating and drawing braids. The length of an Artin word equals the number of crossings seen in a braid diagram. Minimum crossing numbers provide a measure of complexity for braids. This paper presents an algorithm for N=3. A three-dimensional configuration space for 3-braids will also be defined and analysed.