Invariants of Classical Braids Valued in $G_{n}^{2}$
  The aim of the present note is to construct invariants of the Artin braid group valued in $G_{N}^{2}$, and further study of groups related to $G_{n}^{3}$. In the groups $G_{n}^{2}$, the word problem is solved; these groups are much simpler than $G_{n}^{3}$. 
Vassily Olegovich Manturov
math.GT