On groups , braids and Brunnian braids.

In [V. O. Manturov, ], the second author defined the -free braid group with strands . These groups appear naturally as groups describing dynamical systems of particles in some 'general position'. Moreover, in [V. O. Manturov and I. M. Nikonov, J. Knot Theory Ramification 24 (2015) 1541009] the second author and Nikonov showed that is closely related to classical braids. The authors showed that there are homomorphisms from the pure braids group on strands to and and they defined homomorphisms from to the free products of . That is, there are invariants for pure free braids by and . On the other hand in [D. A. Fedoseev and V. O. Manturov, J. Knot Theory Ramification 24(13) (2015) 1541005, 12 pages] Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning - points on strands. In [S. Kim, arXiv:submit/1548032], the first author studied with parity and points. the author constructed a homomorphism from to the group with parity. In the present paper, we investigate the groups and extract new powerful invariants of classical braids from . In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids. [ABSTRACT FROM AUTHOR]