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 :

\[\vcenter{\xy 0;/r1pc/:
    @={(0.5,-0.5),(0,0)}, @@{*{\vtwistneg}},(0,-2),\vtwist,
    @i@={(0,-1),(2,0),(2,-2)} @@{="save";"save"-(0,1),**@{-}}\endxy}
    \quad = \quad
    \vcenter{\xy 0;/r1pc/:
    @={(0.5,0),(0,-0.5),(0.5,-1)}, @@{*{\vtwistneg}},
    @i@={(0,0),(2,-1),(0,-2)} @@{="save";"save"-(0,1),**@{-}}\endxy}\]

And the desired result enter image description here


2 Answers 2


Update (2019-05-09): With the advent of TikZ's pic syntax I figured the braids package was well overdue an overhaul. The syntax below is for the original version of the package which is now depreciated (though still works). See the package documentation for how to use the new syntax.

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:


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

\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}

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

\pgfkeyssetvalue{/tikz/braid height}{1cm}
\pgfkeyssetvalue{/tikz/braid width}{1cm}
\pgfkeyssetvalue{/tikz/braid start}{(0,0)}
\pgfkeyssetvalue{/tikz/braid colour}{black}

  \@ifnextchar^{\message{Got sup}\cross@sup}{\cross@sub}}




  \foreach \thread in {1,...,\value{strands}}
    \pgfmathsetmacro{\strand@x}{\thread * \braid@w}
    \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);
     \draw[braid] \pgfkeysvalueof{/tikz/braid start} ++(\strand@x pt,\cross@y pt) -- ++(0,-\braid@h);

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

  \pgfkeysgetvalue{/tikz/braid width}{\braid@w}
  \pgfkeysgetvalue{/tikz/braid height}{\braid@h}

\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}

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).)

  • 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.
    – PHL
    Apr 29, 2011 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.
    – PHL
    Apr 29, 2011 at 17:45
  • @Matsaya: Sure. I could add it to the list at: meta.tex.stackexchange.com/q/1220/86 Apr 29, 2011 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. Apr 29, 2011 at 20:55
  • 1
    very nice work.
    – PHL
    May 10, 2011 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.

  • 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.
    – PHL
    Apr 29, 2011 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. Apr 29, 2011 at 13:57
  • Is it normal that the compilation with metapost gives me an error ? Anyway, it works fine with metafun.
    – PHL
    Apr 29, 2011 at 21:04

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