  This paper concerns connections between dynamical systems, knots and helicity
of vector fields. For a divergence-free vector field on a closed $3$-manifold
that generates an Anosov flow, we show that the helicity of the vector field
may be recovered as the limit of appropriately weighted averages of linking
numbers of periodic orbits, regarded as knots. This complements a classical
result of Arnold and Vogel that, when the manifold is a real homology
$3$-sphere, the helicity may be obtained as the limit of the normalised linking
numbers of typical pairs of long trajectories. We also obtain results on the
asymptotic distribution of weighted averages of null-homologous periodic
orbits.
