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arXiv 1606.03563math.GT
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在组$G_{n}^{k}$、辫子和Brunnian辫子上

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

S. Kim, V. O. Manturov

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14 pages, 6 figures
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摘要：

在\cite{Manturov}中，第二作者用$n$链$G_{n}^{k}$定义了无$k$编织群。这些群自然地表现为描述处于某种“一般位置”的$n$粒子的动力学系统的群。此外，在\cite{ManturovNikonov}中，第二作者和I.M.Nikonov表明，$G_{n}^{k}$与经典编织物密切相关。作者证明了从$n$链上的纯辫群到$G_{n}^{3}$和$G_{n}^{4}$存在同态，并定义了从$G_{n}^{k}$到$\mathbb{Z}_{2}$的自由积的同态.也就是说，对于$G_{n}^{3}$和$G_{n}^{4}$的纯自由编织物，存在不变量。另一方面，在\cite{FedoseevManturov} D. A. Fedoseev和第二作者研究了经典的辫子，此外结构：奇偶校验和点上的每一股。作者证明了奇偶性，这是一个抽象的结构，具有几何意义-点链。在\cite{Kim}中，第一作者用奇偶和点研究了$G_{n}^{2}$。构造了一个从$G_{n+1}^{2}$到群$G_{n}^{2}$的具有奇偶性的同态.本文研究了群$G_{n}^{3}$，并从$G_{n}^{3}$中提取了经典辫子的新的强不变量。特别地，这些不变量允许人们区分布伦辫子的非平凡性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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