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Embedding calculus and Vassiliev spectral sequence

Syunji Moriya

TL;DR

The paper proves that for the space of long knots in R^3, the Vassiliev and Sinha spectral sequences have isomorphic E_infty pages over a field, with a defined bidegree shift; together with known Sinha degeneracy, this yields Vassiliev degeneration at E_1 over Q for the diagonal and non-diagonal parts. A Thom-space model based on fat diagonals is developed to bridge embedding-calculus data with Vassiliev resolutions, and a detailed zigzag of spaces (PK, C, T, and U_i) connects the punctured-knot picture to the Thom-space picture, enabling a stable isomorphism of spectral sequences. The approach yields partial computations of unstable differentials and clarifies when weight-system totalizations capture finite-type invariants (via degeneracy conditions). The work also proposes a naturality conjecture linking the embedding-calculus tower and finite-type invariants and outlines how a Grothendieck-Teichmüller–type action could control low-degree weight analyses, suggesting broad implications for the topology of knot spaces and related algebraic structures.

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 $\leq n$ if and only if the parts of the Sinha sequence of bidegree $(-i,i)$ degenerate at $E_2$ for $i\leq 2n$. 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.

Embedding calculus and Vassiliev spectral sequence

TL;DR

The paper proves that for the space of long knots in R^3, the Vassiliev and Sinha spectral sequences have isomorphic E_infty pages over a field, with a defined bidegree shift; together with known Sinha degeneracy, this yields Vassiliev degeneration at E_1 over Q for the diagonal and non-diagonal parts. A Thom-space model based on fat diagonals is developed to bridge embedding-calculus data with Vassiliev resolutions, and a detailed zigzag of spaces (PK, C, T, and U_i) connects the punctured-knot picture to the Thom-space picture, enabling a stable isomorphism of spectral sequences. The approach yields partial computations of unstable differentials and clarifies when weight-system totalizations capture finite-type invariants (via degeneracy conditions). The work also proposes a naturality conjecture linking the embedding-calculus tower and finite-type invariants and outlines how a Grothendieck-Teichmüller–type action could control low-degree weight analyses, suggesting broad implications for the topology of knot spaces and related algebraic structures.

Abstract

Vassiliev spectral sequence and Sinha spectral sequence are both related to cohomology of the space of long knots . Although they have different origins, the Vassiliev -page and the Sinha -page are isomorphic (up to a degree shift). In this paper, we prove that they have isomorphic -pages if the coefficient ring is a field. Together with degeneracy of the Sinha sequence, this implies that the Vassiliev sequence degenerates at -page over including the non-diagonal part. Our result also implies that for any coefficient field, the space of finite type knot invariants is isomorphic to the space of weight systems of weight if and only if the parts of the Sinha sequence of bidegree degenerate at for . 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.

Paper Structure

This paper contains 22 sections, 40 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Preliminary
  4. Punctured knot model $\mathcal{PK}$ and cosimplicial model $\mathcal{S}^\bullet$
  5. The functors $\mathcal{F}$, $\check{\mathsf{C}}$ and total complex
  6. Vassiliev spectral sequence
  7. Translation of the punctured knot model
  8. Configuration space model $\mathcal{C}$
  9. Thom space model $\mathcal{T}$
  10. Equivalence between $\mathcal{PK}$ and $\mathcal{T}$
  11. A resolution of $\mathcal{T}$
  12. The evaluation map
  13. Outline of the construction
  14. Definition of $U_1$ and $\psi:\bar{\Sigma}\to U_1$
  15. Basic properties of $U_1$ and $\psi$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mathsf{k}$ be a field.

Figures (3)

  • Figure 1: the map $e_{P}$ for $P=\{\{01\},\{23\},\{4\},\{5\}\}$ ($p_0=(-1,0,0)$)
  • Figure 2: definition of $(y^{st},w^{st})$
  • Figure 3: discontinuity of the sliding operation

Theorems & Definitions (112)

  • Theorem 1.1
  • Corollary 1.2: kontsevich1 for the diagonal part
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: sinhasinha1
  • Definition 2.6
  • ...and 102 more