  In a previous work [11], the author considered a representation of the braid
group \rho: B_n\to GL_m(\Bbb Z[q^{\pm 1},t^{\pm 1}]) (m=n(n-1)/2), and proved
it to be faithful for n=4. Bigelow [3] then proved the same representation to
be faithful for all n by a beautiful topological argument. The present paper
gives a different proof of the faithfulness for all n. We establish a relation
between the Charney length in the braid group and exponents of t. A certain
B_n-invariant subset of the module is constructed whose properties resemble
those of convex cones. We relate line segments in this set with the Thurston
normal form of a braid.
