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<h1>A braid applet
<font size=2>version 1.0 (June 2007)</font></h1>
<h4>by 
<a href="http://www.math.unicaen.fr/~dehornoy">Patrick Dehornoy</a>
(conception, text) and
<a href="http://www.math.unicaen.fr/~jfroment">Jean Fromentin</a>
(realization, implementation)
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<b>What does this applet?</b>

It allows you to draw braid diagrams, and to perform a few basic operations, namely recognizing if a braid diagram is really braided, or, more generally, if a braid diagram can be continuously deformed into another one. The program grew out from an applet by 
<a href="http://math.ucsb.edu/~bigelow/">Stephen Bigelow</a>. 
We intend to gradually enrich it with more braid operations.
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<applet code="HandleReduction.class" height="640" width="480">
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<b>How to use the applet?</b>

Click on the buttons at the bottom of the window; at the moment, three procedures are implemented -- click on colored words to obtain explanation about technical terms.

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<font color="red">Draw</font>: You enter a braid word, i.e., any word in the letters 'a', 'A', 'b', 'B', ..., 'z', 'Z', and the program draws the braid diagram <a href="AppletInfo.html">encoded by</a> that word.
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<font color="red">Reduce</font>: You enter a braid word as above, the program draws the corresponding braid diagram, but, then, it dynamically transforms it so as to obtain a simpler diagram, for instance a diagram with no crossing if this is possible; in other words, the program decides if the initial diagram is really braided, i.e., it solves the <a href="AppletInfo.html">Braid Triviality Problem</a>; the method used to transform the diagram is <a href="AppletInfo.html">handle reduction</a>, one of the many ways to recognize trivial braid diagrams. 
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<font color="red">Compare</font>: You enter two braid words, the program draws the corresponding braid diagrams <i><b>D, D'</b></i> and, then, it decides if one can deform <i><b>D</b></i> into <i><b>D'</b></i> continuously, i.e., it solves the <a href="AppletInfo.html">Braid Isotopy Problem</a>; the method consists in building a new diagram
<i><b>D''</b></i> by stacking the mirror image of <i><b>D</b></i> over <i><b>D'</b></i> and then  applying <a href="AppletInfo.html">handle reduction</a> to <i><b>D''</b></i>.

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<font color="red">Controls</font> enables you to change the parameters of animations:
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<li><font color="red">Speed</font> adjusts the speed of diagram deformations (it mostly depends on the machine on which the applet runs);

<li><font color="red">Mode</font> selects between <font color="red">Continuous</font> (deformations occur automatically) and <font color="red">Step-by-step</font> (deformations occur when you click on the <font color="red">Reduce</font> button).

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<A href="mailto:dehornoy@math.unicaen.fr,jean.fromentin@math.unicaen.fr">Send a question or a comment</A>

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