# zbMATH — the first resource for mathematics

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
Full Text: