A Fox-Neuwirth Basis for the Sinha Spectral Sequence

TL;DR

This paper introduces a combinatorial framework using Fox-Neuwirth trees to describe the Sinha spectral sequence, which approximates the (co)homology of long knots in R^m, enhancing understanding of its structure across dimensions.

Contribution

It establishes an equivalent cosimplicial structure on a CW complex with cells indexed by Fox-Neuwirth trees, providing a combinatorial presentation of the Sinha spectral sequence for all m≥2.

Findings

Provides a combinatorial presentation of the spectral sequence.

Establishes an equivalent cosimplicial structure on CW complexes.

Applies to all dimensions m≥2 with various coefficients.

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 \`a 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.