On groups G_n^k, braids and Brunnian braids

2016/06/11 by Kim, S., Manturov, V. O.
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1606.03563

Abstract

In \citeManturov the second author defined the k-free braid group with n strands G_n^k. These groups appear naturally as groups describing dynamical systems of n particles in some "general position". Moreover, in \citeManturovNikonov the second author and I.M.Nikonov showed that G_n^k is closely related classical braids. The authors showed that there are homomorphisms from the pure braids group on n strands to G_n³ and G_n⁴ and they defined homomorphisms from G_n^k to the free product of ℤ₂. That is, there are invariants for pure free braids by G_n³ and G_n⁴. On the other hand in \citeFedoseevManturov 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 \citeKim, the first author studied G_n² with parity and points. the author construct a homomorphism from G_n+1² to the group G_n² with parity. In the present paper, we investigate the groups G_n³ and extract new powerful invariants of classical braids from G_n³. In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.

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