Embedding calculus and Vassiliev spectral sequence

Vassiliev spectral sequence and Sinha spectral sequence are both related to cohomology of the space of long knots R→ R³. Although they have different origins, the Vassiliev E₁-page and the Sinha E₂-page are isomorphic (up to a degree shift). In this paper, we prove that they have isomorphic E_∞-pages if the coefficient ring is a field. Together with degeneracy of the Sinha sequence, this implies that the Vassiliev sequence degenerates at E₁-page over Q including the non-diagonal part. Our result also implies that for any coefficient field, the space of finite type n knot invariants is isomorphic to the space of weight systems of weight ≤ n if and only if the parts of the Sinha sequence of bidegree $(-i,i)$ degenerate at E₂ for i≤ 2n. For the construction of the isomorphism, we use a variant of Thom space model which was introduced in the author's previous paper and captures embedding calculus of the knot space in terms of fat diagonals. As a byproduct of the construction, we give a partial computation on differentials of unstable versions of the Vassiliev sequence which converge to finite dimensional approximations of the knot space.