  In this paper we study the reduction curves of a braid, and how they can be
used to decompose the braid into simpler ones in a precise way, which does not
correspond exactly to the decomposition given by Thurston theory. Then we study
how a cyclic sliding (which is a particular kind of conjugation) affects the
normal form of a braid with respect to the normal forms of its components.
Finally, using the above methods, we provide the example of a family of braids
whose sets of sliding circuits (hence ultra summit sets) have exponential size
with respect to the number of strands and also with respect to the canonical
length.
