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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

Author

Kim, S.

Published

2016

Language

EN