Weak faithfulness properties for the Burau representation
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''.
scientific article; zbMATH DE number 896587
scientific article