2025/09/28 by Syunji Moriya, Moriya, Syunji
Mathematics · Physics and Astronomy · #55R80 #55U10 #57R40 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum chaos and dynamical systems #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.23766
openalex publication_date 2025/09/28 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
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.