Weak faithfulness properties for the Burau representation
type: Item
instance of: scholarly article
title: Weak faithfulness properties for the Burau representation
zbMATH Open document ID: 0859.20028
DOI: 10.1016/0166-8641(95)00078-X
author: Patrick Dehornoy
published in: Topology and its Applications
publication date: 1997-04-10T00:00:00Z
review text: Let \(\rho_n:B_n\to\text{GL}(n,\mathbb{Z}[t,t^{-1}])\) be the Burau representation of the Artin braid group \(B_n\) with standard generators \(\sigma_1,\dots,\sigma_{n-1}\), and consider the property \(P_k(B_n)\) stating that \(\rho_n(\beta)\neq 1\) for every \(\beta\in B_n\) representable as a word with at most \(k\) occurrences of \(\sigma_1\) and with no occurrence of \(\sigma^{-1}_1\). It is known that \(\rho_n\) is faithful if and only if \(P_k(B_n)\) holds for all \(k\). In this paper, the author shows results implying \(P_k(B_3)\) (\(\forall k\)), \(P_3(B_n)\) (\(n\leq 7\)), \(P_4(B_4)\) and disproving \(P_8(B_6)\), \(P_5(B_7)\) to ``measure the degree of faithfulness of'' \(\rho_n\) via the properties \(P_k(B_n)\) and their variants. Some quadratic relations for the Burau matrices \(\rho_n(B_n)\) are also shown by classifying all (Hermitian) matrices with respect to which the Burau matrices are ``unitary''.
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 896587
zbMATH Keywords: Burau representation
zbMATH Keywords: Artin braid groups
zbMATH Keywords: Burau matrices
reviewed by: Hiroaki Nakamura
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 896587