Braids and Handle Reduction

Applet implemented by Ester Dalvit (01/02/2010). Click here if it does not work
This is a new version of Jean Fromentin's applet. It illustrates the handle reduction algorithm by Patrick Dehornoy.


A braid is drawn. With the left mouse button you can rotate it; use the scroll to zoom on the image.
Using the buttons you can add crossings at the bottom of the braid, go back to the original braid or clear the braid to obtain a single strand.
The play/pause button starts or stop the handle reduction algorithm.

What happens?

A braid is made of some strands which are fixed at the top and bottom ends. To decide if two given braids are “equal” we can move the strands of the first one to try and obtain the same configuration of the second braid. If we succeed in this process, then the two braids are “equal”, but if this is not the case, we are not sure that they are different. It may happen that they are in fact the same braid, but we did not choose the right movements to show it.
Deciding whether a braid can be transformed into the trivial braid is not always easy. For example, if you look at the braid in the applet, it is not soon clear that it is the trivial braid.
The handle reduction algorithm is a process that gives an answer to the latter question. It says how you have to move the strands of a given braid (as shown in the applet when play is clicked): the braid is trivial if and only if the applet stops when all strands are vertical.
Click here to know what a handle is and what moves the algorithm performs.


Zip archive containing the java source code. To compile it you need to install java and JoGL.
Jar archive containing the applet.

This project was partially supported by ANR TheoGar.
Thanks to Patrick Dehornoy for the conception of the applet, Jean Fromentin for the collaboration and Paolo Bellingeri for the invitation to the Laboratoire de Mathématiques Nicolas Oresme in Caen.

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Contact me at:
Department of Mathematics of the University of Trento
Laboratory of Didactics and Communication of Mathematics.