The group $G_{n}^{2}$ with a parity and with points

Abstract

In~\cite{Ma} Manturov studied groups $G_{n}^{k}$ for fixed integers $n$ and $k$ such that $k<n$. In particular, $G_{n}^{2}$ is isomorphic to the group of free braids of $n$-stands. In~\cite{KiMa} Manturov and the author studied an invariant valued in free groups not only for free braids but also for free tangles, which is derived from the group $G_{n}^{2}$. On the other hands, in~\cite{FeMa} Manturov and Fedoseev studied groups $Br_{2}^{n}$ of virtual braids with parity and groups $Br_{d}^{n}$ of virtual braids with dots. They showed that there is a monomorphism from $Br_{2}^{n}$ to $Br_{d}^{n}$ and it is deduced that a parity of the braid can be represented by a geometric object, dots on strands. In this paper we study $G_{n}^{2}$ with structures, which are corresponded to parity and points on a braid, which are denoted by $G_{n,p}^{2}$ and $G_{n,d}^{2}$, respectively. In section 3, it is proved that there is a monomorphism from $G_{n}^{2}$ to $G_{n,p}^{2}$ and that there is a monomorphism from $G_{n,p}^{2}$ to $G_{n,d}^{2}$. By the homomorphism from $G_{n}^{2}$ to $G_{n,p}^{2}$, it can be deduced that a given parity of a braid has geometric representation, which is the number of points on the braid. In section 4, it can be proved that for each element $β$ in $G_{n,d}^{2}$, an element in $G_{n+1}^{2}$ obtained by adding another strand by tracing points on $β$. That is, a parity of free braid of $n$-stands is represented not only by points on strands, but also by an $n+1$-th strand. Conversely, for a braid of $n+1$-strands, a braid of $n$-strands is obtained by deleting one strand of the braid of $n+1$-strand. Finally, we will simply discuss about the way to adjust the previous observations to know whether a given braid is Brunnian or not.