A simplification in Vassiliev's spectral sequence

Abstract

We establish a simple formula to compute the incidence signs in the spectral sequence used by Vassiliev to define finite-type cohomology classes in the space of long knots. We consider only the subcomplex of tree diagrams, which carries all the cohomology in low degrees and is the most difficult to compute. On the way, it is shown that by getting rid of the most complicated ingredients in the boundary maps, one obtains an isomorphic complex.