Complete description of Agol cycles of pseudo-Anosov 3-braids

The equivalence class of an Agol cycle is a conjugacy invariant of a pseudo-Anosov map. Mosher defined train tracks in the torus associated to Farey intervals and investigated the relation between the train tracks and the continued fraction expansions of quadratic irrational numbers. We study Mosher’s train tracks and describe Agol cycles of all the pseudo-Anosov $3$-braids.