The rational homology of spaces of long knots in codimension<i>&gt;</i>2

Abstract

We determine the rational homology of the space of long knots in R d for d 4. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E 1 page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree d 1, which can be obtained as the homology of an explicit graph complex and is in theory completely computable.