The Groups $G_{n}^{2}$ with Additional Structures

In the paper [1], V. O. Manturov introduced the groups $G_{n}^{k}$ depending on two natural parameters $n>k$ and naturally related to topology and to the theory of dynamical systems. The group $G_{n}^{2}$, which is the simplest part of $G_{n}^{k}$, is isomorphic to the group of pure free braids on $n$ strands. In the present paper, we study the groups $G_{n}^{2}$ supplied with additional structures – parity and points; these groups are denoted by $G_{n,p}^{2}$ and $G_{n,d}^{2}$. First, we define the groups $G_{n,p}^{2}$ and $G_{n,d}^{2}$, then study the relationship between the groups $G_{n}^{2}$, $G_{n,p}^{2}$, and $G_{n,d}^{2}$. Finally, we give an example of a braid on $n+1$ strands, which is not the trivial braid on $n+1$ strands, by using a braid on $n$ strands with parity. After that, the author discusses links in $S_{g} \times S^{1}$ that can determine diagrams with points; these points correspond to the factor $S^{1}$ in the product $S_{g} \times S^{1}$.