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).