Embedding calculus and Vassiliev spectral sequence

Abstract

Vassiliev spectral sequence and Sinha spectral sequence are both related to cohomology of the space of long knots $\mathbb{R}\to {\mathbb{R}}^3$. Although they have different origins, the Vassiliev $E_1$-page and the Sinha $E_2$-page are isomorphic (up to a degree shift). In this paper, we prove that they have isomorphic $E_{\infty }$-pages if the coefficient ring is a field. Together with degeneracy of the Sinha sequence, this implies that the Vassiliev sequence degenerates at $E_1$-page over $\mathbb{Q}$ 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 $\le n$ if and only if the parts of the Sinha sequence of bidegree $(-i,i)$ degenerate at $E_2$ for $i\le 2n$. 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.

48 pages, 3 figures, v2: Modified Corollary 1.4 (1) in v1 since its proof included an error. Removed (2) of the corollary as it follows from known diagonal collapse. Modified Conjecture 7.4. Also corrected minor errors