Galois symmetries of knot spaces

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 $(n+1)$ th Goodwillie–Weiss approximation is a $p$ -local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p + 1$ .