  Given a flow on a 3-dimensional integral homology sphere, we give a formula
for the Euler characteristic of its transverse surfaces, in terms of boundary
data only. We illustrate the formula with several examples, in particular with
surfaces of low genus. As an application, we show that for a right-handed flow
with an ergodic invariant measure, the genus is an asymptotic invariant of
order 2 proportional to the helicity.
