  We suggest a new algorithm for finding a canonical representative of a given
braid, and also for the harder problem of finding a $\sigma_1$-consistent
representative. We conjecture that the algorithm is quadratic-time. We present
numerical evidence for this conjecture, and prove two results: (1) The
algorithm terminates in finite time. (2) The conjecture holds in the special
case of 3-string braids - in fact, we prove that the algorithm finds a
minimal-lenght representative for any 3-string braid.
