The Agol cycle is a complete invariant of a pseudo-Anosov map. In this talk I will show what can an Agol cycle tell about pseudo-Anosov 3-braids. I will show topological aspect as well as number theoretic aspect of the Agol cycle. In particular, I will relate an Agol cycle with a periodic arc sequence in the Farey tessellation. This is joint work with my student, Elaina Aceves.