Related: mathoverflow.net/q/442622/137265

Thanks for the reference, it clarified things a little. However, as Alex pointed out, I'm looking for a somewhat-specific example! (in the paper, I'm working with sutures only annuli, and in the balanced case) Unfortunately, they do not deal with such examples!

Regarding my previous comment: be careful that the webpage I linked doesn't talk about the (only) obstruction to this algorithm, which is mentioned in Dehornoy's paper (see page 6, paragraph below figure 1.4). Implementing it directly without this precaution makes the recursive pile loop endleslessly!

Because links tend to break after a few years, here is an explanation of the algorithm for direct computer implementation: matematita.science.unitn.it/applet/handlereduction/en/info.h‌​tml.

I must say, I had tried the argument without much success, so thank you very much for detailing it !