Determinant of the crossing matrix of a braid

2025/09/10 by Ayaka Shimizu, Shimizu, Ayaka
Engineering · #Geotechnical Engineering and Soil Stabilization

paper · pdf · doi:10.48550/arxiv.2509.08464

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.

Citations

• The CN matrix of a pure braid projection
• Characterization of the OU matrix of a braid diagram
• Determinant of the OU matrix of a braid diagram

Related

• Determinant of the OU matrix of a braid diagram
• Crossing Matrices of Positive Braids
• The CN matrix of a pure braid projection
• Characterization of the OU matrix of a braid diagram
• The crossing matrix and the extended first Johnson homomorphism of a braid group