The CN matrix of a pure braid projection

Yuko Ozawa; Ayaka Shimizu; Yoshiro Yaguchi

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arXiv · 2506.08659

The CN matrix of a pure braid projection

Yuko Ozawa·Ayaka Shimizu·Yoshiro Yaguchi

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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.

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Yuko Ozawa, Ayaka Shimizu, Yoshiro Yaguchi. 2025-06-10. The CN matrix of a pure braid projection. https://arxiv.org/abs/2506.08659

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