On groups $G_{n}^{k}$, braids and Brunnian braids

TL;DR

This paper explores the groups $G_{n}^{k}$ related to braids, introduces new invariants for classical braids derived from $G_{n}^{3}$, and demonstrates their effectiveness in identifying non-trivial Brunnian braids.

Contribution

It introduces novel invariants for classical braids based on the groups $G_{n}^{3}$, enhancing the ability to distinguish non-trivial Brunnian braids.

Findings

New invariants for classical braids derived from $G_{n}^{3}$.

Invariants can detect non-triviality of Brunnian braids.

Connections established between $G_{n}^{k}$ groups and classical braid properties.

Abstract

In \cite{Manturov} 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 \cite{ManturovNikonov} 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 $G_{n}^{3}$ and $G_{n}^{4}$ and they defined homomorphisms from $G_{n}^{k}$ to the free product of $\mathbb{Z}_{2}$. That is, there are invariants for pure free braids by $G_{n}^{3}$ and $G_{n}^{4}$. On the other hand in \cite{FedoseevManturov} 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 \cite{Kim}, the first author studied $G_{n}^{2}$ with parity and points. the author construct a homomorphism from $G_{n+1}^{2}$ to the group $G_{n}^{2}$ with parity. In the present paper, we investigate the groups $G_{n}^{3}$ and extract new powerful invariants of classical braids from $G_{n}^{3}$. In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.