  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.
