The Sinha spectral sequence for long knot cohomology is isomorphic, from the first page on, to a spectral sequence built from Fox-Neuwirth trees, for every dimension and coefficient group.
Non collapse of the Sinha spectral sequence for knots in R^3
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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.
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A Fox-Neuwirth Basis for the Sinha Spectral Sequence
The Sinha spectral sequence for long knot cohomology is isomorphic, from the first page on, to a spectral sequence built from Fox-Neuwirth trees, for every dimension and coefficient group.