I've tried to google how to generate words in two generators up to conjugacy (this means I only want one representative for each conjugacy class). Sadly, I come up with articles that have, in the introduction, things like "It is well known that the elements of the free group of rank two can be enumerated by the rationals".
I think I understood that there is a bijection between the conjugacy classes of elements of the free group on two generators and the rational numbers, and this bijection has something to do with the continued fraction associated with a given rational number. I couldn't find any reference on the topic though, so if anyone could give me one I would be grateful.
I would also appreciate any other method to compute the conjugacy classes of the free group on two generators. (I need an algorithm for a program I am writing)