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The braid group \(B_4\) is linear. (English) Zbl 0988.20023
The author studies a linear representation \(\rho\colon B_n\to\text{GL}_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])\) (where \(m=n(n-1)/2\)). Instead of the usual Artin presentation of \(B_n\), the author uses a new presentation with a new generating subset \(Q\) and formulates a relation for the Charney length function with respect to \(Q\). He formulates a conjecture implying that: (a) \(\rho\) is faithful, (b) the Charney length function with respect to \(Q\) satisfies the relation. The author proves this conjecture for \(n=4\).

MSC:
20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
57M07 Topological methods in group theory
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