Determinant of the crossing matrix of a braid

Q: What type of study is this?

This is a Quantitative Study study.

Ayaka Shimizu

doi:10.48550/arxiv.2509.08464

September 10, 2025Open Access

Determinant of the crossing matrix of a braid

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Key Points

The purified determinant of a braid demonstrates that P(b1 b2) equals P(b2 b1), ensuring braid invariance.
The study defines a braid invariant as the purified determinant P(b), derived from the crossing matrix of pure braids.
These findings demonstrate key properties of the purified determinant as it relates to the algebraic structure of braids.
Implications of the results may extend to broader studies in topology and algebraic geometry, highlighting braid properties.

Implication

This analysis reveals a new braid invariant indicating the determinant of crossing matrices for pure braids.

Abstract

In this paper, we define a braid invariant, the purified determinant $P(b)$ of a braid b, considering the determinant of the crossing matrix of a pure braid derived from b, and show that P(b₁ b₂)=P(b₂ b₁) for any pair of n-braids b₁ and b₂.

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Cite This Study

Ayaka Shimizu (2025) studied this question.

synapsesocial.com/papers/68ed1896f29694dd1da78c13 https://doi.org/10.48550/arxiv.2509.08464

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Determinant of the crossing matrix of a braid

Key Points

Implication

Abstract

Discussion

Cite This Study

Determinant of the crossing matrix of a braid

Key Points

Implication

Abstract

Discussion

Cite This Study

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