  We exploit the Galois symmetries of the little disks operads to show that
many differentials in the Goodwillie-Weiss spectral sequences approximating the
homology and homotopy of knot spaces vanish at a prime $p$. Combined with
recent results on the relationship between embedding calculus and finite-type
theory, we deduce that the $(n+1)$-st Goodwillie-Weiss approximation is a
$p$-local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p
+ 1$.

Algebraic Topology