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

TL;DR

This paper explores the algebraic structures of groups related to free braids, introducing new variants with parity and points, and demonstrates their geometric and combinatorial representations, including monomorphisms and strand manipulations.

Contribution

It introduces new groups $G_{n,p}^{2}$ and $G_{n,d}^{2}$ with structures for parity and points, and establishes monomorphisms and geometric interpretations connecting these groups to free braids.

Findings

Existence of monomorphisms between $G_{n}^{2}$, $G_{n,p}^{2}$, and $G_{n,d}^{2}$

Geometric representation of braid parity via points and strands

Method to determine if a braid is Brunnian

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 $\beta$ in $G_{n,d}^{2}$, an element in $G_{n+1}^{2}$ obtained by adding another strand by tracing points on $\beta$. 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.