An even symmetric 6x6 matrix is the crossing-count matrix of a pure 6-braid projection exactly when it satisfies the T0 condition.
Kawauchi, Lectures on knot theory (in Japanese), Kyoritsu Shuppan Co
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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.