Abelian quotients for groups generated by cubed half-twists

Below we construct non-cyclic and torsion-free abelian quotients for subgroups of braid groups generated by cube powers of half-twists. In the case of 3 and 4 strands we compute the abelianization of these groups. Also, we get abelianizations of level-3 congruence subgroups of braid groups on 3 and 4 strands. Moreover, we construct non-cyclic and torsion-free abelian quotients for level-3 congruence subgroups of braid groups. The constructions are based on the original construction of Dennis Johnson \cite{DJ}, who computed an abelian quotient of the Torelli group.