We obtain asymptotics for the average value taken by a Vassiliev invariant on knots appearing as periodic orbits of an Axiom A flow on
$S^3.$ The methods used also give asymptotics for the writhe of periodic orbits. Our results are analogous to those of G. Contreras [Average linking numbers of closed orbits of hyperbolic flows. J. Lond. Math. Soc. (2) 51 (1995), 614–624] for average linking numbers.