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This question led to a new package:
braids — Blog: The Braids Package

I search to draw braids diagram, if possible in xy-pic. I already know how to do that with the knot module and some low-level instructions and I search a nicer way to do that.

The actual code :

\documentclass{minimal}
\usepackage[all,knot]{xy}
\begin{document}
\[\vcenter{\xy 0;/r1pc/:
    @={(0.5,-0.5),(0,0)}, @@{*{\vtwistneg}},(0,-2),\vtwist,
    @i@={(0,-1),(2,0),(2,-2)} @@{="save";"save"-(0,1),**@{-}}\endxy}
    \quad = \quad
    \vcenter{\xy 0;/r1pc/:
    @={(0.5,0),(0,-0.5),(0.5,-1)}, @@{*{\vtwistneg}},
    @i@={(0,0),(2,-1),(0,-2)} @@{="save";"save"-(0,1),**@{-}}\endxy}\]
\end{document}

And the desired result enter image description here

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2 Answers 2

up vote 20 down vote accepted

Update (2011-09-06): The resulting package is now on CTAN as http://www.ctan.org/pkg/braids and you can read about it on our blog at http://tex.blogoverflow.com/2011/09/the-braids-package/.


Taking up Charles' challenge, here's a way of doing this in TikZ/PGF. First, the output:

braid

Next, the user code; namely, if the yucky preamble were in a nice package, this is what you would type:

\begin{document}
\begin{tikzpicture}
\braid[braid colour=red,strands=3,braid start={(0,0)}]
{\sigma_1 \sigma_2 \sigma_1^{-1}}
\node[font=\Huge] at (4.5,-1.5) {\(=\)};
\braid[strands=3,braid start={(5,0)}]
{\sigma_2 \sigma_1 \sigma_2}
\end{tikzpicture}
\end{document}

Lastly, the yucky preamble (actually including the user code so you can just copy this for a MWE):

\documentclass{standalone}
\usepackage{tikz}
\newcounter{braid}
\newcounter{strands}
\pgfkeyssetvalue{/tikz/braid height}{1cm}
\pgfkeyssetvalue{/tikz/braid width}{1cm}
\pgfkeyssetvalue{/tikz/braid start}{(0,0)}
\pgfkeyssetvalue{/tikz/braid colour}{black}
\pgfkeys{/tikz/strands/.code={\setcounter{strands}{#1}}}

\makeatletter
\def\cross{%
  \@ifnextchar^{\message{Got sup}\cross@sup}{\cross@sub}}

\def\cross@sup^#1_#2{\render@cross{#2}{#1}}

\def\cross@sub_#1{\@ifnextchar^{\cross@@sub{#1}}{\render@cross{#1}{1}}}

\def\cross@@sub#1^#2{\render@cross{#1}{#2}}


\def\render@cross#1#2{
  \def\strand{#1}
  \def\crossing{#2}
  \pgfmathsetmacro{\cross@y}{-\value{braid}*\braid@h}
  \pgfmathtruncatemacro{\nextstrand}{#1+1}
  \foreach \thread in {1,...,\value{strands}}
  {
    \pgfmathsetmacro{\strand@x}{\thread * \braid@w}
    \ifnum\thread=\strand
    \pgfmathsetmacro{\over@x}{\strand * \braid@w + .5*(1 - \crossing) * \braid@w}
    \pgfmathsetmacro{\under@x}{\strand * \braid@w + .5*(1 + \crossing) * \braid@w}
    \draw[braid] \pgfkeysvalueof{/tikz/braid start} +(\under@x pt,\cross@y pt) to[out=-90,in=90] +(\over@x pt,\cross@y pt -\braid@h);
    \draw[braid] \pgfkeysvalueof{/tikz/braid start} +(\over@x pt,\cross@y pt) to[out=-90,in=90] +(\under@x pt,\cross@y pt -\braid@h);
    \else
    \ifnum\thread=\nextstrand
    \else
     \draw[braid] \pgfkeysvalueof{/tikz/braid start} ++(\strand@x pt,\cross@y pt) -- ++(0,-\braid@h);
    \fi
   \fi
  }
  \stepcounter{braid}
}

\tikzset{braid/.style={double=\pgfkeysvalueof{/tikz/braid colour},double distance=1pt,line width=2pt,white}}

\newcommand{\braid}[2][]{%
  \begingroup
  \pgfkeys{/tikz/strands=2}
  \tikzset{#1}
  \pgfkeysgetvalue{/tikz/braid width}{\braid@w}
  \pgfkeysgetvalue{/tikz/braid height}{\braid@h}
  \setcounter{braid}{0}
  \let\sigma=\cross
  #2
  \endgroup
}
\makeatother


\begin{document}
\begin{tikzpicture}
\braid[braid colour=red,strands=3,braid start={(0,0)}]%
{\sigma_1 \sigma_2 \sigma_1^{-1}}
\node[font=\Huge] at (4.5,-1.5) {\(=\)};
\braid[strands=3,braid start={(5,0)}]
{\sigma_2 \sigma_1 \sigma_2}
\end{tikzpicture}
\end{document}

Improvements: the actual drawing commands could be optimised a little. The braids don't actually start at the start, but are shifted one over (not hard to correct). It should also be possible to work out the number of strands from the given braid: if we encounter an element using a strand we've yet to see, simply draw it from the top to the current height.

(I did try to do a bit of catcode tomfoolery so that one could use s in place of \sigma, but I couldn't get it to work (and, yes, I did replace the command by an environment in trying to do this).)

share|improve this answer
    
Thanks a lot, it's perfect even it's not a xy-pic solution. I wait a little more before accept your answer to see if there is a xy guru around here. –  Matsaya Apr 29 '11 at 12:58
    
Could I suggest you to make your code a package and to release it on Ctan as it could be useful for (a lot of) people. –  Matsaya Apr 29 '11 at 17:45
    
@Matsaya: Sure. I could add it to the list at: meta.tex.stackexchange.com/q/1220/86 –  Loop Space Apr 29 '11 at 17:59
    
@Matsaya: Thinking about this a bit more, may I ask what you'd want from such a package? I'm quite new to knots, links, and braids so am not sure what functionality would be useful. –  Loop Space Apr 29 '11 at 20:55
1  
very nice work. –  Matsaya May 10 '11 at 20:53

Stijn Symens has a nice Metapost package for drawing braids based on the abstraction of giving a string that specifies the crossings in the order that they occur, using lower case letters to draw left over right crossings and upper case the converse. He has written a brief introduction.

I don't know of any comparatively concise way of representing braids in Xypic, or PGF/Tikz for that matter, though writing a similar package for PGF should not be so hard for a PGF guru.

share|improve this answer
    
And writing similar module for xy should not be sor hard for a xy guru. But it seems that there are more PGF gurus here. –  Matsaya Apr 29 '11 at 12:56
    
@Matsaya: Right, but PGF has abstractions that make this sort of thing easier: look at how Andrew's soln uses \foreach \thread in {1,...,\value{strands}} and the \pgfkeys... macros. Metapost is easiest, because it has proper string primitives. –  Charles Stewart Apr 29 '11 at 13:57
    
Is it normal that the compilation with metapost gives me an error ? Anyway, it works fine with metafun. –  Matsaya Apr 29 '11 at 21:04

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