The CN matrix of a pure braid projection

TL;DR

This paper characterizes the CN matrix patterns of pure braid projections, especially for 6-braids, and explores related matrices like the OU matrix and crossing matrix for positive pure braids.

Contribution

It provides a detailed characterization of CN matrices for pure 6-braids and links these to other crossing-related matrices, advancing understanding of braid projection structures.

Findings

Characterization of CN matrices for pure 6-braids

Identification of patterns in CN matrices

Relations between CN, OU, and crossing matrices

Abstract

The CN matrix of an $n$-braid projection $B$ is an $n \times n$ matrix such that each $(i,j)$ entry indicates the number of crossings between $i^{th}$ and $j^{th}$ strands of $B$. In this paper, several patterns of an $n \times n$ matrix to be a CN matrix are discussed, and the CN matrix of a pure 6-braid projection is characterized. As an application, the OU matrix of a pure 6-braid diagram and the crossing matrix of a positive pure 6-braid are also characterized.