# The Root Extraction Problem for Generic Braids

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Garside Structure of ${\mathbb{B}}_{n}$

- The partial order ≼ in G defined by $a\preccurlyeq b$ if ${a}^{-1}b\in \mathcal{P}$ is a lattice order. If $a\preccurlyeq b$ we say that a is a prefix of b. The lattice structure implies that for all $a,b\in G$ there exists a unique meet $a\wedge b$ and a unique join $a\vee b$ with respect to ≼. Notice that this partial order is invariant under left-multiplication.
- The set of simple elements $\mathcal{S}\text{:}=\{s\in G\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}1\preccurlyeq s\preccurlyeq \Delta \}$ is finite and generates G.
- Conjugation by $\Delta $ preserves $\mathcal{P}$, that is, ${\Delta}^{-1}\mathcal{P}\Delta =\mathcal{P}$.
- $\mathcal{P}$ is atomic: the atoms are the indivisible elements of $\mathcal{P}$ (elements $a\in \mathcal{P}$ for which there is no decomposition $a=bc$ with non-trivial elements $b,c\in \mathcal{P}$). Then, for every $x\in \mathcal{P}$ there is an upper bound on the number of atoms in a decomposition of the form $x={a}_{1}{a}_{2}\cdots {a}_{n}$, where each ${a}_{i}$ is an atom.

#### 2.2. Normal Forms, Cyclings and Decyclings

**Definition**

**1**

**.**The left normal form of an element $x\in {\mathbb{B}}_{n}$ is the unique decomposition $x={\Delta}^{p}{x}_{1}\cdots {x}_{l}$ so that $p\in \mathbb{Z}$, $l\ge 0$, ${x}_{i}\in \mathcal{S}\backslash \{1,\Delta \}$ for $i=1,\dots ,l$, and ${x}_{i}{x}_{i+1}$ is a left weighted decomposition, for $i=1,\dots ,l-1$.

**Definition**

**2.**

**Definition**

**3**

**.**Let $x={\Delta}^{p}{x}_{1}\cdots {x}_{l}$ be in left normal form, with $l>0$. The cycling and decycling of x are the conjugates of x defined, respectively, as

**not**the left normal form of $\mathbf{c}\left(x\right)$. So $\mathbf{c}\left(x\right)$ could a priori have a shorter normal form (with less factors). A similar situation happens for $\mathbf{d}\left(x\right)$.

**Definition**

**4.**

#### 2.3. Summit Sets

**Definition**

**5**

**.**Given $x\in {\mathbb{B}}_{n}$, the cyclic sliding of x is defined as $\mathfrak{s}\left(x\right)=\mathfrak{p}{\left(x\right)}^{-1}x\phantom{\rule{0.222222em}{0ex}}\mathfrak{p}\left(x\right)$, where $\mathfrak{p}\left(x\right)=\iota \left(x\right)\wedge \partial \left(\phi \right(x\left)\right)$.

**Theorem**

**1**

**.**Given $x\in {\mathbb{B}}_{n}$, there are integers $0\le k<t$ such that ${\mathfrak{s}}^{k}\left(x\right)={\mathfrak{s}}^{t}\left(x\right)$. For every such pair of integers, one has ${\mathfrak{s}}^{k}\left(x\right)\in USS\left(x\right)$.

**Theorem**

**2**

**.**Let $x\in {\mathbb{B}}_{n}$ and suppose that x is conjugate to a rigid braid. Then there is an integer $k>0$ such that ${\mathfrak{s}}^{k}\left(x\right)$ is rigid. Moreover, the conjugating element α from x to ${\mathfrak{s}}^{k}\left(x\right)$, that is,

**Definition**

**6**

**.**Let $x\in {\mathbb{B}}_{n}$ and $y\in USS\left(x\right)$. A simple non-trivial element $s\in \mathcal{S}$ is said to be a minimal simple elementfor y if ${y}^{s}\in USS\left(x\right)$ and ${y}^{t}\notin USS\left(x\right)$, for every $1\prec t\prec s$.

**Lemma**

**1**

**.**Let $y\in USS\left(x\right)$ with $\ell \left(y\right)>0$ and let s be a minimal simple element for y. Then, s is a prefix of either $\iota \left(y\right)$ or $\partial \left(\phi \right(y\left)\right)$, or both.

**Lemma**

**2**

**.**A braid $y\in USS\left(x\right)$ with $\ell \left(y\right)>0$ is rigid if and only if none of the edges starting at y is bi-colored.

**Definition**

**7.**

**Remark**

**1.**

#### 2.4. Generic Braids

**Theorem**

**3**

**.**The proportion of braids in $B\left(r\right)$ whose ultra summit set is minimal tends to 1 exponentially fast, as r tends to infinity.

**Theorem**

**4**

**.**The proportion of braids x in $B\left(r\right)$ which are conjugate to a rigid braid $y={\alpha}^{-1}x\alpha $, in such a way that α is a positive braid with $\ell \left(\alpha \right)<\ell \left(x\right)$, tends to 1 exponentially fast, as r tends to infinity.

**Definition**

**8.**

**Theorem**

**5**

**.**Let $x\in {\mathbb{B}}_{n}$ and $y\in USS\left(x\right)$. Let $PC\left(y\right)={p}_{1}\cdots {p}_{t}$ as above. If $USS\left(x\right)$ is minimal, then all elements in $USS\left(x\right)$ are rigid, $Z\left(x\right)\simeq Z\left(y\right)\simeq {\mathbb{Z}}^{2}$, and one of the following conditions holds:

- (i)
- $USS\left(x\right)$has two orbits under cycling, conjugate to each other by Δ, and $Z\left(y\right)=\langle {\Delta}^{2},PC\left(y\right)\rangle $.
- (ii)
- $USS\left(x\right)$has one orbit under cycling, conjugate to itself by Δ, and:
- -
- If$\tau \left(y\right)=y$, then$Z\left(y\right)=\langle \Delta ,PC\left(y\right)\rangle $.
- -
- If$\tau \left(y\right)\ne y$, then t is even and$Z\left(y\right)=\langle {\Delta}^{2},\phantom{\rule{0.222222em}{0ex}}{p}_{1}\cdots {p}_{\frac{t}{2}}{\Delta}^{-1}\rangle $.

## 3. k-th Root Problem

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 4. An Algorithm to Find the k-th Root of a Braid

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

Algorithm 1: ind a k-th root of a braid x. |

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Cumplido, M.; González-Meneses, J.; Silvero, M.
The Root Extraction Problem for Generic Braids. *Symmetry* **2019**, *11*, 1327.
https://doi.org/10.3390/sym11111327

**AMA Style**

Cumplido M, González-Meneses J, Silvero M.
The Root Extraction Problem for Generic Braids. *Symmetry*. 2019; 11(11):1327.
https://doi.org/10.3390/sym11111327

**Chicago/Turabian Style**

Cumplido, María, Juan González-Meneses, and Marithania Silvero.
2019. "The Root Extraction Problem for Generic Braids" *Symmetry* 11, no. 11: 1327.
https://doi.org/10.3390/sym11111327