A braid applet version 1.1 (October 2007)

by Patrick Dehornoy (conception, text) and Jean Fromentin (realization, implementation)

What does this applet? This applet demonstrates some aspects of the mathematical theory of braids. 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 Stephen Bigelow. Thanks to Marc Autord for his help.

How to use the applet? 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.
  • Draw: 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 encoded by that word.

  • Reduce: 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 Braid Triviality Problem; the method used to transform the diagram is handle reduction, one of the many ways to recognize trivial braid diagrams.

  • Compare: You enter two braid words, the program draws the corresponding braid diagrams D, D' and, then, it decides if one can deform D into D' continuously, i.e., it solves the Braid Isotopy Problem; the method consists in building a new diagram D'' by stacking the mirror image of D over D' and then applying handle reduction to D''.

  • Controls enables you to change the parameters of animations:
    • Speed adjusts the speed of diagram deformations (it mostly depends on the machine on which the applet runs);
    • Mode selects between Continuous (deformations occur automatically) and Step-by-step (deformations occur when you click on the Reduce button).

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