Matrix presentations of braids and applications

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries − 1 , 0 , -1, 0, or 1 1 and the free monoid generated by 2 n 2n generators σ 1 , ⋯ , σ n , σ 1 − 1 , ⋯ , σ n − 1 \sigma _{1}, \cdots ,\sigma _{n}, \sigma _{1}^{-1},\cdots , \sigma _{n}^{-1} and relation σ i ± 1 σ j ± 1 = σ j ± 1 σ i ± 1   ( | i − j | ≥ 2 ) \sigma _{i}^{\pm 1}\sigma _{j}^{\pm 1} = \sigma _{j}^{\pm 1}\sigma _{i}^{\pm 1}~ (|i-j| \geq 2) and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.