An even symmetric 6x6 matrix is the crossing-count matrix of a pure 6-braid projection exactly when it satisfies the T0 condition.
Characterization of the OU matrix of a braid diagram
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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.
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The CN matrix of a pure braid projection
An even symmetric 6x6 matrix is the crossing-count matrix of a pure 6-braid projection exactly when it satisfies the T0 condition.