On the Homology of Spaces of Long Knots

Abstract

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.

NONTRIVIAL CLASSES IN H*(Imb(S1,ℝn)) FROM NONTRIVALENT GRAPH COCYCLES

“…A different approach to the problem of computing the cohomology of Imb (S 1 , R n ) has been considered by V. Vassiliev in [11]: there exists in fact a spectral sequence converging to the cohomology of Imb (S 1 , R n ) for every n ≥ 4, whose E 1 term coincides with the graph cohomology (see also [10]). Thus, Vassiliev spectral sequence gives an "upper bound" to the cohomology of Imb (S 1 , R n ), since nontrivial cohomology classes of Imb (S 1 , R n ) must come from some element in E 1 , while [4] gives a "lower bound", since at least those elements corresponding to trivalent diagrams give rise to nontrivial elements in cohomology.…”

Poisson structures on the homology of the space of knots

“…See also the pictures in Budney's paper [2]. The reason why we may regard • i and µ 2 as respectively 'insertion' and 'concatenation' can be found in the definition of Poisson algebras operad H * (X n ) (see Turchin [17,18]). …”

“…When moreover the cosimplicial space arises from the operad O with multiplication, then its E 1 -term is the Hochschild complex of the homology operad H * (O). It is known (Gerstenhaber-Voronov [6], Turchin [17,18]) that there exist a natural product and a bracket on such a complex which induce the Gerstenhaber algebra structure, the degree one Poisson algebra structure, on the homology. Note that the Gerstenhaber algebra structure also comes from the action of the chains of the little disks operad (Deligne's conjecture; see McClure-Smith [13]).…”

“…We regard λ as a Poisson algebra structure on the homology (see Cohen [5] • is an operad with multiplication, there exist a natural product and a bracket on E 1 -term (see Gerstenhaber-Voronov [6], Turchin [17,18]) that make the E 2 -term HH * (O • ) a Gerstenhaber algebra, that is, a Poisson algebra of degree one [17,18]. This Poisson structure defined on E 2 -term in fact descends on further terms (Proposition 4.8).…”

“…By unraveling the descriptions of the Hochschild complex (O , ∂) (see Gerstenhaber-Voronov [6], Turchin [17,18]), we can see the following. In our case Ψ is defined on E 1 and makes E 2 a Poisson algebra.…”