Characterization of the OU matrix of a braid diagram

TL;DR

This paper characterizes the OU matrix of pure braid diagrams and crossing matrices of positive pure braids for up to 5 strands, providing standard forms and insights into their structure.

Contribution

It introduces a comprehensive characterization of OU matrices and crossing matrices for braids with up to 5 strands, including standard forms for general braids.

Findings

OU matrix of pure braids characterized for up to 5 strands

Crossing matrix of positive pure braids characterized for up to 5 strands

Standard forms of OU matrices provided for general braids

Abstract

The OU matrix of a braid diagram is a square matrix that represents the number of over/under crossings of each pair of strands. In this paper, the OU matrix of a pure braid diagram is characterized for up to 5 strands. As an application, the crossing matrix of a positive pure braid is also characterized for up to 5 strands. Moreover, a standard form of the OU matrix is given and characterized for general braids of up to 5 strands.