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

In [8], Manturov studied groups $G_{n}^{k}$ given by the group presentation, from which many invariants for knots and dynamical systems are derived. In [10], V.O.Manturov and I.M.Nikonov found that pure braids groups are embedded in $G_{n}^{k}$ and constructed invariants valued in free product of cyclic groups. Tangles are significant generalization of braids and we have the following question : is it possible to generalize invariants, which are asserted, to the case of tangles. By $G_{n}^{2}$ analogously tangles can be studied. In [6], the invariants valued in free groups are extended to the case of free tangles. The groups $G_{n}^{2}$ which are simplest case of the $G_{n}^{k}$ are known as free braid groups of $n$-strands, which are known as other names, see [1] and [2].

On the other hands, in [3] V.O.Manturov and D.A.Fedoseev studied virtual braids group $Br_{2}^{n}$ with parity and virtual braids group $Br_{d}^{n}$ with dots. They showed that there is a homomorphism $h$ from $Br_{2}^{n}$ to $Br_{d}^{2}$. And it was shown that $Br_{2}^{n}$ is homomorphically embedded in $Br_{d}^{n}$ by showing the following statement:

if two braids with parity are equivalent as braids with dots, then they are equivalent as braids with parity.

It means that the parity of braids has geometric meaning; the number of points on braids.

In this paper, analogously we will generalizethe groups $G_{n}^{2}$ with structures; parity and points, which are denoted by $G_{n,p}^{2}$ and $G_{n,d}^{2}$, respectively. We are interested in the following question:
two objects are equivalent in a lager category, then are they equivalent as themselves?
Which means that a category is embedded in the larger category. For example, the following statements are related with the above problem.

1. (1)

   if two classical knots are equivalent as virtual knots, then they are equivalent as classical knots.

2. (2)

   if two classical braids are equivalent as virtual briads, then they are equivalent as classical briads.

In [5] the first statement is proved and the second statement is proved in [4].

As the above question, we showed the following two statements:

1. (1)

   if two element in $G_{n}^{2}$ are equivalent in $G_{n,p}^{2}$, then they are equivalent in $G_{n}^{2}$.

2. (2)

   if two element in $G_{n,p}^{2}$ are equivalent in $G_{n,d}^{2}$, then they are equivalent in $G_{n,p}^{2}$.

From the above statements, the followings are deduced: Since $G_{n}^{2}$ is isomorphic to group of free braids with $n$ strands, parity can be generalized to the case of free braids with respect to $G_{n,p}^{2}$. Moreover, parity is defined abstractly, but it can be geometrically presented by points. On the other hands point on a braid of $n$ strands means ’liking’ between $(n+1)$-th strand and others. Moreover points on the braid of $n$-strands can be considered as traces of $n+1$-th strand and a homomorphism from $G_{n,d}^{2}$ to $G_{n+1}^{2}$ can be defined by adding a new strand by ‘following the traces’(points) of it. By the above observations, we can find a homomorphism from $G_{n,p}^{2}$ to $G_{n+1}^{2}$ and, roughly speaking, each value of parity of crossings of the braid of $n$-strands can be represented by the number of ‘linking’ between $n+1$-strand and the other two strands, in which a crossing is. Conversely a braid with points can be obtained from a braid of $n+1$ strands by deleting one strand and placing points on places, on which deleted stand was.And a braid with parity can be obtained from a braid with points by counting the number of points. Then two braids can be distinguished by braids with parity. In section 2, we simply remind basic definitions and simple results. In section 3, we consider the following question:
two objects are equivalent in a larger category, then are they equivalent as themselves?
in the case of $G_{n}^{2}$, $G_{n,p}$ and $G_{n,d}^{2}$. In section 4, relations between $G_{n,d}^{2}$ and $G_{n+1}$ are studied. And we will show geometric meanings and examples of braids of $n+1$ strand, which are distinguished from trivial braids by braids with parity.

Firstly, we recall basic definitions and introduce simple results.

In [9] Manturov showed that colored (pure) free braids can be presented by elements in $G_{n}^{2}$. In fact, the relations for $G_{n}^{2}$ are related with Reidemeister moves for free braids.

A colored free braid diagram is a free braid diagram such that each component is enumerated. If it is a pure free braid diagram, we may assume that the enumeration agrees with $x$-terms of $1$-valent vertices of the strands.

This moves are related with relations of $G_{n}^{2}$ in order. Now we construct a function from the set $PB_{n}$ of colored pure free braids with $n$ components to $G_{n}^{2}$. Let $\{\epsilon_{k}\}_{k=1}^{m}\subset(0,1)$ be a set of values such that a $4$-valent vertex is placed in $\mathbb{R}\times\{\epsilon_{k}\}$ and $\epsilon_{k}<\epsilon_{k+1}$ for each $k=1,\cdots m-1$. Without loss of generality, we assume that there is the only one vertex on $\mathbb{R}\times\{\epsilon_{k}\}$ for each $k$. Let $c_{k}$ be the $4$-valent vertex on $\mathbb{R}\times\{\epsilon_{k}\}$. Define a function $\iota$ from $PB_{n}$ to $G_{n}^{2}$ as the following: For each vertex $c_{k}$, if it contains edges in $i_{k}$-th and $j_{k}$-th components, then give a generator $a_{i_{k}j_{k}}$ in $G_{n}^{2}$. Define $\iota(\beta)=a_{i_{1}j_{1}}\cdots a_{i_{m}j_{m}}$, see Fig. 2.

From the discovery of parity by Manturov [7], invariants for classical knots are extended to invariants for virtual knots. In [3] Fedoseev and Manturov studied classical braids with structures, which are related to parity and dots on strands of classical braids. With those structures, parity can be extended to the case of braids and generalized parity can be deduced by braids with dots. The group of braids with parity and with points are defined as follows:

In the group $Br_{n}^{d}$, each generator $\tau_{i}$ is presented by a point on $i$-th strand of the braid. Now we define a mapping $f:Br_{n}^{2}\rightarrow Br_{n}^{d}$ by

$f(\sigma_{i}^{\epsilon})=\left\{\begin{array}[]{cc}\sigma_{i}&\text{if}~\epsilon=0,\\ \tau_{i}\sigma_{i}\tau_{i}&\text{if}~\epsilon=1.\\ \end{array}\right.$

In [3] they proved that this mapping is injective homomorphism. Moreover the following proposition can be proved.

In the following sections, we study $G_{n}^{2}$ with structures which are related to parity and points algebraically.

In this section, we get enhancements of $G_{n}^{2}$ by adding some structures, which are related to parity and points on strands of colored pure braids. Firstly, we define a group $G_{n,p}^{2}$ and we call it $G_{n}^{2}$ with parity.

The relations of $G_{n,p}^{2}$ are closely related to the axioms of parity [7]. We call $a_{ij}^{0}$($a_{ij}^{1}$) an even generator(an odd generator). If a word $\beta$ in $G_{n,p}^{2}$ has no odd generators, then we call $\beta$ an even word (or an even element). $G_{n}^{2}$ can be considered as a subgroup of $G_{n,p}^{2}$ by a homomorphism $i$ from $G_{n}^{2}$ to $G_{n,p}^{2}$ defined by $i(a_{ij})=a_{ij}^{0}$. It is easy to show that $i$ is well-defined. Moreover, the following statement can be proved.

Now, we define a group $G_{n,d}^{2}$ and we call it $G_{n}^{2}$ with points.

We call $\tau_{i}$ a generator for a point on $i$-th component or simply, a point on $i$-th component. Geometrically, $a_{ij}$ is corresponded to a 4-valent vertex and $\tau_{i}$ is corresponded to a point on the $i$-th strand of the free braid, see Fig. 3.

In fact, two groups $G_{n,p}^{2}$ and $G_{n,d}^{2}$ are closely related to each other. Define a function $\phi$ from $G_{n,p}^{2}$ to $G_{n,d}^{2}$ by

$\phi(a_{ij}^{\epsilon})=\left\{\begin{array}[]{cc}a_{ij}&\text{if}~\epsilon=0,\\ \tau_{i}a_{ij}\tau_{i}&\text{if}~\epsilon=1.\\ \end{array}\right.$

Geometrically, the image of $a_{ij}^{1}$ is a crossing between $i$-th and $j$-th strands of the braid with two points just before and after of the crossing on $i$-th strand, see Fig. 4.

Notice that, since we can get $\tau_{i}a_{ij}\tau_{i}=\tau_{j}a_{ij}\tau_{j}$ from the relation $\tau_{i}\tau_{j}a_{ij}\tau_{j}\tau_{i}=a_{ij}$, two diagrams with point in Fig. 4 are equivalent. On the other hand, the number of $\tau_{i}$ and $\tau_{j}$ before a given crossing $\phi(a_{ij}^{\epsilon})$ is equal to $\epsilon$ modulo $2$, because every image of $a_{kl}^{\epsilon}$ before $\phi(a_{ij}^{\epsilon})$ has two points or no points.

Let $H_{n,d}^{2}=\{\beta\in G_{n,d}^{2}~|~N_{i}(\beta)=0~\text{mod}~2,~i\in\{1,\cdots,n\}\}$, where $N_{i}(\beta)$ is the number of $\tau_{i}$ in $\beta$. Notice that, by the definition of $\phi$, it is clear that the image of $G_{n,p}^{2}$ by $\phi$ goes into the set $H_{n,d}^{2}$. Since every relation of $G_{n,d}^{2}$ preserves the number of $\tau$’s modulo $2$, $H_{n,d}^{2}$ is a subgroup of $G_{n,d}^{2}$.