I have read a statement from Sossinsky and Prasolov' s book "Knots, Lİnks, Braids and 3Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced form, page 54. My claim is that this statement is not true: take $b_2b_1b_2^{1}b_3^{1}b_3^{1}$ and $b_3^{1}b_2b_3b_2b_1b_2^{1}b_2^{1}b_3^{1}b_2^{1}$. They represent isotopic braids but they are not the same. What is the correct form of such a characterization?

Here "reduced" refer to the socalled handle reduction algorithm introduced by Dehornoy. So far I remember, this algorithm does not provide a normal form, therefore I agree that the statement is false: two reduced word may represent the same braid even if they are different. Indeed, it can be checked with the following Java applet: http://www.math.unicaen.fr/~tressapp/index.html (it applies handle reduction the other way as you does, i.e. it pushes handles to the left, so in that case the reduction will produce the same word. But if you swap $b_1$ and $b_3$ you still get isotopic braid, and now the applet will tell you that the words are already reduced). The true statement is that a word represent the trivial braid if and only if its reduction is the empty word. Of course it still solve the word problem: checking if $w_1,w_2$ represent the same braid is the same as checking if $w_1w_2^{1}$ represents the trivial braid. Edit. For the sake of completeness let me recall why not leading to a normal form is not a negative point against this algorithm.



For a very nice (if slightly typesettingchallenged) survey of this subject, see Patrick Dehornoy's survey. 

