# Involutory Quandles and Dichromatic Links

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts and Terminology

**Definition**

**1**

- 1.
- $x\u25b9x=x$, for all $x\in X$.
- 2.
- $(x\u25b9y)\u25b9y=x$, for all $x,y\in X$.
- 3.
- $(x\u25b9y)\u25b9z=(x\u25b9z)\u25b9(y\u25b9z)$, for all $x,y,z\in X$.

**Definition**

**2**

**Definition**

**3**

- Any non-empty set X with operation $x\u22b3y=x$, for all $x,y\in X$ is a kei. It is called the trivial kei.
- Let $\langle \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\rangle :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to \mathbb{R}$ be a symmetric bi-linear form on ${\mathbb{R}}^{n}$. Let X be the subset of ${\mathbb{R}}^{n}$ consisting of vectors $\overrightarrow{u}$ such that $\langle \overrightarrow{u},\overrightarrow{u}\rangle \ne 0$. Then, the operation$$\overrightarrow{u}\u22b3\overrightarrow{v}=\frac{2\langle \overrightarrow{u},\overrightarrow{v}\rangle}{\langle \overrightarrow{u},\overrightarrow{u}\rangle}\overrightarrow{v}-\overrightarrow{u}$$
- A set $X=\mathbb{Z}$ with operation $x\u22b3y=2y-x$, for all $x,y\in \mathbb{Z}$ is a kei.
- A group $X=G$ with operation $x\u22b3y=y{x}^{-1}y$ is a kei. It is called the core kei of the group G.

## 3. Construction of Dikei

**Definition**

**4.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Example**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Applications

**Example**

**3.**

**Justification:**

**Justification:**

**Example**

**4.**

**Justification:**

**Justification:**

**Example**

**5.**

**Justification:**

**Justification:**

**Example**

**6.**

**Justification:**

**Justification:**

**Example**

**7.**

**Justification:**

**Justification:**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Bataineh, K.; Saidi, I.
Involutory Quandles and Dichromatic Links. *Symmetry* **2020**, *12*, 111.
https://doi.org/10.3390/sym12010111

**AMA Style**

Bataineh K, Saidi I.
Involutory Quandles and Dichromatic Links. *Symmetry*. 2020; 12(1):111.
https://doi.org/10.3390/sym12010111

**Chicago/Turabian Style**

Bataineh, Khaled, and Ilham Saidi.
2020. "Involutory Quandles and Dichromatic Links" *Symmetry* 12, no. 1: 111.
https://doi.org/10.3390/sym12010111