The CN matrix of a pure braid projection

The CN matrix of an n-braid projection B is an [Formula: see text] matrix such that each [Formula: see text] entry indicates the number of crossings between ith and jth strands of B. In this paper, several patterns of an [Formula: see text] 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.