Examples of Anosov flows with genus one Birkhoff sections

We show that a transitive Anosov flow with orientable stable and unstable foliations that either (i) admits a Birkhoff section whose first return map is a Penner type pseudo-Anosov map, or (ii) is totally periodic admits a genus one Birkhoff section. This provides evidence for a conjecture of Fried and Ghys. The proof utilizes a result of the author on the horizontal Goodman surgery operation. To apply this result for showing (i), we establish correspondence between horizontal Goodman surgery on pseudo-Anosov flows and horizontal surgery on veering triangulations in the layered setting.