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A Fox-Neuwirth Basis for the Sinha Spectral Sequence

Andrea Marino

TL;DR

This work addresses the problem of efficiently modeling the Sinha spectral sequence for the space of long knots by replacing Kontsevich spaces with a Fox-Neuwirth–based cosimplicial framework indexed by Fox-Neuwirth trees.It constructs a barycentric Fox-Neuwirth bicomplex and proves an isomorphism, at the level of $E^r$-pages, to the Sinha spectral sequence for all $m\ge 2$ and coefficients $A$, unifying combinatorial and geometric approaches.A zig-zag of semicosimplicial homotopy equivalences ties together the weighted hairy tree model $\mathrm{WHT}_m$, the sd\,BZ_m model, and Konts_m, enabling transfer of spectral-sequence results across these models, including rational formality and collapse phenomena.The framework provides practical computational leverage for configuration-space cohomology in knot theory, offering a combinatorial, geometrically meaningful route to study invariants and their differentials across dimensions.

Abstract

Recently, Sinha defined a spectral sequence approximating the (co)homology of the space of long knots in R^m modulo immersions, stemming from a cosimplicial structure on the compactified configuration spaces à la Kontsevich. We provide an equivalent cosimplicial structure on (the barycentric subdivision of) a regular CW complex with cells indexed by Fox-Neuwirth trees. As a corollary, we give a combinatorial presentation of the Sinha Spectral Sequence in terms of Fox-Neuwirth trees for all dimensions m>=2 and all coefficients.

A Fox-Neuwirth Basis for the Sinha Spectral Sequence

TL;DR

This work addresses the problem of efficiently modeling the Sinha spectral sequence for the space of long knots by replacing Kontsevich spaces with a Fox-Neuwirth–based cosimplicial framework indexed by Fox-Neuwirth trees.It constructs a barycentric Fox-Neuwirth bicomplex and proves an isomorphism, at the level of $E^r$-pages, to the Sinha spectral sequence for all $m\ge 2$ and coefficients $A$, unifying combinatorial and geometric approaches.A zig-zag of semicosimplicial homotopy equivalences ties together the weighted hairy tree model $\mathrm{WHT}_m$, the sd\,BZ_m model, and Konts_m, enabling transfer of spectral-sequence results across these models, including rational formality and collapse phenomena.The framework provides practical computational leverage for configuration-space cohomology in knot theory, offering a combinatorial, geometrically meaningful route to study invariants and their differentials across dimensions.

Abstract

Recently, Sinha defined a spectral sequence approximating the (co)homology of the space of long knots in R^m modulo immersions, stemming from a cosimplicial structure on the compactified configuration spaces à la Kontsevich. We provide an equivalent cosimplicial structure on (the barycentric subdivision of) a regular CW complex with cells indexed by Fox-Neuwirth trees. As a corollary, we give a combinatorial presentation of the Sinha Spectral Sequence in terms of Fox-Neuwirth trees for all dimensions m>=2 and all coefficients.

Paper Structure

This paper contains 25 sections, 48 theorems, 252 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. The Fox-Neuwirth Stratification on Configuration Spaces
  4. The Cosimplicial Structure on Kontsevich spaces
  5. Barycentric Fox-Neuwirth Bicomplex
  6. Combinatorics of cosimplicial Fox-Neuwirth trees
  7. The shape-tree lemma
  8. Relationship between positions and labels
  9. Spaces of Weighted Trees
  10. Twisted geometric realization
  11. Contravariance conditions
  12. Non-degenerate realization
  13. Positively Weighted Trees
  14. Construction by twisted geometric realization
  15. Equivalence with $\textrm{sd\,}\mathrm{BZ}_m$
  16. ...and 10 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 $mn - \sum a_i$. The images of the $\textrm{Conf}(\Gamma)$ are the interiors of cells in an equivariant CW structure on the one-point compactification $\textrm{Conf}_n(\

Figures (18)

  • 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: The McClure-Smith cosimplicial structure on a multiplicative operad
  • Figure 5: The shape of a Fox-Neuwirth tree after several cosimplicial moves
  • ...and 13 more figures

Theorems & Definitions (142)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 3.1: Shape-Tree
  • Remark 3.2
  • ...and 132 more