Crossing matrices and Thurstonʹs normal form for braids

In this paper we study the relation between positive braids, their canonical normal forms and their crossing matrices. We characterize positive braids of canonical length at most 2 by their crossing matrices, but prove that for length 3 or larger this is no longer true. We characterize matrices which can be crossing matrices of braids with canonical length at most 2, answering a question by Elrifai and Morton.