\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{braid group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{geometric_definition}{Geometric definition}\dotfill \pageref*{geometric_definition} \linebreak
\noindent\hyperlink{grouptheoretic_definition}{Group-theoretic definition}\dotfill \pageref*{grouptheoretic_definition} \linebreak
\noindent\hyperlink{artin_presentation}{Artin presentation}\dotfill \pageref*{artin_presentation} \linebreak
\noindent\hyperlink{in_terms_of_automorphisms_on_free_groups}{In terms of automorphisms on free groups}\dotfill \pageref*{in_terms_of_automorphisms_on_free_groups} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_moduli_space_of_monopoles}{Relation to moduli space of monopoles}\dotfill \pageref*{relation_to_moduli_space_of_monopoles} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{surface_braid_groups}{Surface braid groups}\dotfill \pageref*{surface_braid_groups} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_moduli_space_of_monopoles_2}{Relation to moduli space of monopoles}\dotfill \pageref*{relation_to_moduli_space_of_monopoles_2} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{braid group} $Br_n$ is the [[group]] whose elements are [[isotopy]] classes of $n$ [[dimension|1-dimensional]] \emph{braids} running vertically in 3-dimensional [[Cartesian space]], the group operation being their concatenation.

Here a \emph{braid} with $n$ strands is thought of as $n$ pieces of string joining $n$ points at the top of the diagram with $n$-points at the bottom.

[[3-strand-braid-1-SVG]]

(This is a picture of a 3-strand braid.)

We can transform / `isotope' these braid diagrams just as we can transform [[knot diagrams]], again using [[Reidemeister moves]]. The `isotopy' classes of braid diagrams form a group in which the composition is obtained by putting one diagram above another.

The identity consists of $n$ vertical strings, so the inverse is obtained by turning a diagram upside down:

This is the inverse of the first 3-braid we saw.

There are useful [[group presentations]] of the braid groups. We will return later to the interpretation of the [[generators and relations]] in terms of diagrams.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{geometric_definition}{}\subsubsection*{{Geometric definition}}\label{geometric_definition}

Let $C_n \hookrightarrow \mathbb{C}^n$ be the space of [[configuration space of points|configurations of n points]] in the [[complex plane]], whose elements are those [[n-tuples]] $(z_1, \ldots, z_n)$ such that $z_i \neq z_j$ whenever $i \neq j$. The [[symmetric group]] $S_n$ acts on $C_n$ by permuting coordinates. Let $C_n/S_n$ be the orbit space (the space of $n$-element subsets of $\mathbb{C}$ if one likes), and let $[z_1, \ldots, z_n]$ be the image of $(z_1, \ldots, z_n)$ under the quotient $\pi: C_n \to C_n/S_n$. We take $p = (1, 2, \ldots, n)$ as basepoint for $C_n$, and $[p] = [1, 2, \ldots n]$ as basepoint for $C_n/S_n$.

\begin{defn}
\label{}\hypertarget{}{}
The \emph{braid group} $Br_n$ is the [[fundamental group]] $\pi_1(C_n/S_n, [p])$. The \emph{pure braid group} $P_n$ is $\pi_1(C_n, p)$.

\end{defn}
Evidently a braid $\beta$ is represented by a path $\alpha: I \to C_n/S_n$ with $\alpha(0) = [p] = \alpha(1)$. Such a path may be uniquely lifted through the covering projection $\pi: C_n \to C_n/S_n$ to a path $\tilde{\alpha}$ such that $\tilde{\alpha}(0) = p$. The end of the path $\tilde{\alpha}(1)$ has the same underlying subset as $p$ but with coordinates permuted: $\tilde{\alpha}(1) = (\sigma(1), \sigma(2), \ldots, \sigma(n))$. Thus the braid $\beta$ is exhibited by $n$ non-intersecting strands, each one connecting an $i$ to $\sigma(i)$, and we have a map $\beta \mapsto \sigma$ appearing as the quotient map of an exact sequence

\begin{displaymath}
1 \to P_n \to Br_n \to S_n \to 1
\end{displaymath}
which is part of a long exact homotopy sequence corresponding to the fibration $\pi: C_n \to C_n/S_n$.

\hypertarget{grouptheoretic_definition}{}\subsubsection*{{Group-theoretic definition}}\label{grouptheoretic_definition}

\hypertarget{artin_presentation}{}\paragraph*{{Artin presentation}}\label{artin_presentation}

The \textbf{Artin braid group}, $Br_{n+1}$, defined using $n+1$ strands is a [[group]] given by

\begin{itemize}%
\item generators: $y_i$, $i = 1, \ldots, n$;


\item relations:

\begin{itemize}%
\item $r_{i,j} \equiv y_i y_j y_i^{-1} y_j^{-1}$ for $i+1 \lt j$


\item $r_{i,i+1}\equiv y_i y_{i+1} y_i y_{i+1}^{-1} y_i^{-1} y_{i+1}^{-1}$ for $1 \leq i \lt n$.



\end{itemize}


\end{itemize}
\hypertarget{in_terms_of_automorphisms_on_free_groups}{}\paragraph*{{In terms of automorphisms on free groups}}\label{in_terms_of_automorphisms_on_free_groups}

The braid group $B_n$ may be alternatively described as the [[mapping class group]] of a 2-disk $D^2$ with $n$ punctures (call it $X_n$). Meanwhile, the [[fundamental group]] $\pi_1(X_n)$ (with basepoint on the boundary) is a [[free group]] $F_n$ on $n$ generators; the functoriality of $\pi_1$ implies we have an induced homomorphism

\begin{displaymath}
Aut(X_n) \to Aut(\pi_1(X_n)) = Aut(F_n).
\end{displaymath}
If an automorphism $\phi: X_n \to X_n$ is isotopic to the identity, then of course $\pi_1(\phi)$ is trivial, and so the homomorphism factors through the quotient $MCG(X_n) = Aut(X_n)/Aut_0(X)$, so we get a homomorphism

\begin{displaymath}
B_n = MCG(X_n) \to Aut(F_n)
\end{displaymath}
and this turns out to be an injection.

Explicitly, the generator $y_i$ used in the Artin presentation above is mapped to the automorphism $\sigma_i$ on the free group on $n$ generators $x_1, \ldots, x_n$ defined by

\begin{displaymath}
\sigma_i(x_i) = x_{i+1}, \sigma_i(x_{i+1}) = x_{i+1}^{-1} x_i x_{i+1}, \; \else\; \sigma(x_j) = x_j.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_moduli_space_of_monopoles}{}\subsubsection*{{Relation to moduli space of monopoles}}\label{relation_to_moduli_space_of_monopoles}

\begin{prop}
\label{ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids}\hypertarget{ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids}{}
\textbf{([[moduli space of monopoles]] is [[stable weak homotopy equivalence|stably weak homotopy equivalent]] to [[classifying space]] of [[braid group]])}

For $k \in \mathbb{N}$ there is a [[stable weak homotopy equivalence]] between the [[moduli space of k monopoles]] \eqref{ModuliSpaceOfkInstantons} and the [[classifying space]] of the [[braid group]] $Braids_{2k}$ on $2 k$ strands:

\begin{displaymath}
\Sigma^\infty \mathcal{M}_k
  \;\simeq\;
  \Sigma^\infty Braids_{2k}
\end{displaymath}
\end{prop}
(\hyperlink{CohenCohenMannMilgram91}{Cohen-Cohen-Mann-Milgram 91})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The first few examples for low values of $n$:

\begin{example}
\label{}\hypertarget{}{}
By default, $Br_1$ has no generators and no relations, so is trivial.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
By default, $Br_2$ has one generator and no relations, so is infinite cyclic.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
(We will simplify notation writing $u = y_1$, $v = y_2$.)

This then has presentation

\begin{displaymath}
\mathcal{P} = ( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1}).
\end{displaymath}
It is also the `trefoil group', i.e., the fundamental group of the complement of a [[trefoil knot]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Simplifying notation as before, we have generators $u,v,w$ and relations

\begin{itemize}%
\item $r_u \equiv  v w v w^{-1} v^{-1} w^{-1}$,
\item $r_v \equiv  u w u^{-1} w^{-1}$,
\item $r_w \equiv  u v u v^{-1} u^{-1} v^{-1}$.

\end{itemize}
\end{example}
\hypertarget{surface_braid_groups}{}\subsection*{{Surface braid groups}}\label{surface_braid_groups}

In terms of the geometric definition above, it is possible to consider configurations of points on surfaces other than the plane, which gives rise to the more general notion of a \emph{surface braid group}. For example, the \textbf{Hurwitz braid group} (or \textbf{sphere braid group}) comes from considering configurations of points on the [[2-sphere]] $S^2$. Algebraically, the Hurwitz braid group $H_{n+1}$ has all of the generators and relations of the Artin braid group $Br_{n+1}$, plus one additional relation:

\begin{displaymath}
y_1 y_2 \dots y_{n-1} y_n^2 y_{n-1}\dots y_2 y_1
\end{displaymath}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[braid group statistics]]


\item [[braid category]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Classical references are

\begin{itemize}%
\item [[Joan S. Birman]], \emph{Braids, links, and mapping class groups}, Princeton Univ Press, 1974.


\item [[R. H. Fox]], L. Neuwirth, \emph{The braid groups}, Math. Scand. \textbf{10} (1962) pp.119-126. \href{http://www.mscand.dk/article.php?id=1624}{pdf}, \href{http://www.ams.org/mathscinet-getitem?mr=150755}{MR150755}



\end{itemize}
A recent monograph is

\begin{itemize}%
\item C. Kassel, V. Turaev, \emph{Braid Groups} , GTM \textbf{247} Springer Heidelberg 2008.

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia: \emph{\href{http://en.wikipedia.org/wiki/Braid_group}{Braid group}}

\end{itemize}
For orderings of the braid group see

\begin{itemize}%
\item [[Patrick Dehornoy]], \emph{Braid groups and left distributive operations} , Transactions AMS \textbf{345} no.1 (1994) pp.115--150.


\item H. Langmaack, \emph{Verbandstheoretische Einbettung von Klassen unwesentlich verschiedener Ableitungen in die Zopfgruppe} , Computing \textbf{7} no.3-4 (1971) pp.293-310.



\end{itemize}
\hypertarget{relation_to_moduli_space_of_monopoles_2}{}\subsubsection*{{Relation to moduli space of monopoles}}\label{relation_to_moduli_space_of_monopoles_2}

On [[moduli spaces of monopoles]] related to [[braid groups]]:

\begin{itemize}%
\item [[Fred Cohen]], [[Ralph Cohen]], B. M. Mann, R. J. Milgram, \emph{The topology of rational functions and divisors of surfaces}, Acta Math (1991) 166: 163 (\href{https://doi.org/10.1007/BF02398886}{doi:10.1007/BF02398886})


\item [[Ralph Cohen]], John D. S. Jones \emph{Monopoles, braid groups, and the Dirac operator}, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 (\href{https://projecteuclid.org/euclid.cmp/1104254240}{euclid:cmp/1104254240})



\end{itemize}
category: knot theory

[[!redirects braid groups]]

[[!redirects braid]] [[!redirects braids]]

[[!redirects pure braid]] [[!redirects pure braids]]

[[!redirects Artin braid group]] [[!redirects Hurwitz braid group]] [[!redirects surface braid group]]



\end{document}