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Non collapse of the Sinha spectral sequence for knots in R^3

Andrea Marino, Paolo Salvatore

TL;DR

This work provides an explicit, $\mathbb{F}_2$-coefficients description of the Sinha spectral sequence for the space of long knots in $\mathbb{R}^3$ up to its third page, via a 3-truncated multicomplex built from Fox-Neuwirth cells and their barycentric structure. It introduces a detailed, computer-assisted framework to compute higher differentials $D_i$ (notably $d^3$) by encoding distributions with Fox polynomials and enforcing multicomplex identities, ultimately showing a nontrivial differential $d^3$ from a 2-dimensional class in $E^3_{6,8}$ to $E^3_{9,10}$. The finding demonstrates that the mod 2 Sinha spectral sequence does not collapse at page 3 for $m=3$, implying non-formality of $\mathbb{E}_3$ over $\mathbb{F}_2$ and motivating refined techniques beyond rational formality in positive characteristics. The paper also develops a general deformation theorem (the Barycentric Deformation Theorem) that translates a bicomplex description into a tractable truncated multicomplex, providing a concrete computational pipeline for early-stage knot invariant calculations and laying groundwork for further differential analysis in low dimensions.

Abstract

We give an explicit description up to the third page of the Sinha homology mod 2 spectral sequence for the space of long knots in $\mathbb{R}^3$, that is conjecturally equivalent to the Vassiliev spectral sequence. The description arises from a multicomplex structure on the Fox Neuwirth chain complexes for euclidean configuration spaces. A computer assisted calculation reveals a non trivial third page differential from a 2-dimensional class, in contrast to the rational case.

Non collapse of the Sinha spectral sequence for knots in R^3

TL;DR

This work provides an explicit, -coefficients description of the Sinha spectral sequence for the space of long knots in up to its third page, via a 3-truncated multicomplex built from Fox-Neuwirth cells and their barycentric structure. It introduces a detailed, computer-assisted framework to compute higher differentials (notably ) by encoding distributions with Fox polynomials and enforcing multicomplex identities, ultimately showing a nontrivial differential from a 2-dimensional class in to . The finding demonstrates that the mod 2 Sinha spectral sequence does not collapse at page 3 for , implying non-formality of over and motivating refined techniques beyond rational formality in positive characteristics. The paper also develops a general deformation theorem (the Barycentric Deformation Theorem) that translates a bicomplex description into a tractable truncated multicomplex, providing a concrete computational pipeline for early-stage knot invariant calculations and laying groundwork for further differential analysis in low dimensions.

Abstract

We give an explicit description up to the third page of the Sinha homology mod 2 spectral sequence for the space of long knots in , that is conjecturally equivalent to the Vassiliev spectral sequence. The description arises from a multicomplex structure on the Fox Neuwirth chain complexes for euclidean configuration spaces. A computer assisted calculation reveals a non trivial third page differential from a 2-dimensional class, in contrast to the rational case.

Paper Structure

This paper contains 31 sections, 19 theorems, 205 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Configuration spaces and Fox-Neuwirth cells
  4. Cosimplicial structure on the Kontsevich spaces
  5. Spectral Sequences for Knots
  6. The theory of multicomplexes
  7. Barycentric Fox-Neuwirth Structure
  8. Definition of cosimplicial moves
  9. On Fox-Neuwirth posets
  10. On Fox-Neuwirth chains
  11. On subdivided Blagojevic-Ziegler space
  12. Associated bicomplex and multicomplex
  13. The Barycentric Deformation Theorem
  14. Truncated Multicomplexes
  15. Framework and statement of the theorem
  16. ...and 16 more sections

Key Result

Theorem 2.3

For any $\Gamma \in \textrm{FN}^{\le}_m(n)$, the subspace $\textrm{Conf}(\Gamma)$ is homeomorphic to a Euclidean ball of dimension $mn - \sum a_i$, and is a cell of an equivariant CW decomposition of the one-point compactification $\textrm{Conf}_n(\mathbb{R}^m)^+$.

Figures (11)

  • Figure 1: The tree associated to the Fox-Neuwirth cell $3 <_2 1 <_1 2 <_0 5 <_1 4 \in \textrm{FN}^{\le}_3(5)$
  • Figure 2: An illustration of $\textrm{sd\,}\mathrm{BZ}_3(2)$ as a PL-sphere
  • Figure 3: A point in $\textrm{Konts}_2(5)$
  • Figure 4: A sketch of the "lower line" and "upper line" for Sinha Spectral Sequence
  • Figure 5: Coface $d_2$, codegeneracy $s_1$, external cofaces of the cell $3 <_2 1 <_1 2 <_0 5 <_1 4$
  • ...and 6 more figures

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 3.1
  • ...and 63 more