Embedding calculus and grope cobordism of knots

We show that the invariants $ev_n$ of long knots in a $3$-manifold, produced from embedding calculus, are surjective for all $n\geq1$. On one hand, this solves some of the remaining open cases of the connectivity estimates of Goodwillie and Klein, and on the other hand, it confirms one half of the conjecture by Budney, Conant, Scannell and Sinha that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. We actually study long knots in any manifold of dimension at least $3$ and develop a geometric understanding of the layers in the embedding calculus tower and their first non-trivial homotopy groups, given as certain groups of decorated trees. Moreover, in dimension $3$ we give an explicit interpretation of $ev_n$ using capped grope cobordisms and our joint work with Shi and Teichner. The main theorem of the present paper says that the first possibly non-vanishing embedding calculus invariant $ev_n$ of a knot which is grope cobordant to the unknot is precisely the equivalence class of the underlying decorated tree of the grope in the homotopy group of the layer. As a corollary, we give a sufficient condition for the mentioned conjecture to hold over a coefficient group. By recent results of Boavida de Brito and Horel this is fulfilled for the rationals, and for the $p$-adic integers in a range, confirming that the embedding calculus invariants are universal rational additive Vassiliev invariants, factoring configuration space integrals.