On groups Gnk, braids and Brunnian braids

Abstract

In [7] the second author defined the k-free braid group with n strands G k n . These groups appear naturally as groups describing dynamical systems of n particles in some "general position". Moreover, in [10] the second author and I.M.Nikonov showed that G k n is closely related classical braids. The authors showed that there are homomorphisms from the pure braids group on n strands to G 3 n and G 4 n and they defined homomorphisms from G k n to the free product of Z 2 . That is, there are invariants for pure free braids by G 3 n and G 4 n . On the other hand in [6] D.A.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 [4], the first author studied G 2 n with parity and points. the author construct a homomorphism from G 2 n+1 to the group G 2 n with parity.In the present paper, we investigate the groups G 3 n and extract new powerful invariants of classical braids from G 3 n . In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.