2018/05/30 by Mauricio Gutiérrez, Gutierrez, Mauricio, Zbigniew Nitecki +1 · 1 citation
Computer Science · Mathematics · #05B20 #20F36 (Primary) #57M04 (Secondary) #57M25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1805.12189
openalex publication_date 2018/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The crossing matrix of a braid on N strands is the N× N integer matrix with zero diagonal whose i,j entry is the algebraic number (positive minus negative) of crossings by strand i over strand j . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of N× N symmetric integer matrices with zero diagonal--in fact, it represents the abelianization of this subgroup. As a function on the whole N-braid group, it is a derivation defined by the action of the symmetric group on square matrices. The set of all crossing matrices can be described using the natural decomposition of any braid as the product of a pure braid with a `permutation braid' in the sense of Thurston, but the subset of crossing matrices for positive braids is harder to describe. We formulate a finite algorithm which exhibits all positive braids with a given crossing matrix, if any exist, or declares that there are none.