In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots inThe first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interprete the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d − 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings we introduce new homological operations on Hochschild complexes. These new operations are in fact the Dyer-Lashof operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

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Keywords: discriminant of the space of knots, bialgebra of chord diagrams, Hochschild complex, operads of Poisson -Gerstenhaber -Batalin-Vilkovisky algebras.This paper is a more detailed version of [T1], where the first term of the Vassiliev spectral sequence (computing the homology of the space of long knots in R d , d ≥ 3) was described in terms of the Hochschild homology of the Poisson algebras operad for d odd, and of the Gerstenhaber algebras operad for d even. In particular, the bialgebra of chord diagrams arises as some subspace of this homology. The homology in question is the space of characteristic classes for Hochschild cohomology of Poisson (resp. Gerstenhaber) algebras considered as associative algebras. The paper begins with necessary preliminaries on operads.Also we give a simplification of the computations of the first term of the Vassiliev spectral sequence.We do not give proofs of the results.

We give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristic is always a restricted Lie algebra.16E40; 18D50, 55P48, 55S12