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We determine the rational homology of the space of long knots in R^d for$d\geq4$. Our main result is that the Vassiliev spectral sequence computingthis rational homology collapses at the E^1 page. As a corollary we get thatthe homology of long knots (modulo immersions) is the Hochschild homology ofthe Poisson algebras operad with a bracket of degree d-1, which can be obtainedas the homology of an explicit graph complex and is in theory completelycomputable.  Our proof is a combination of a relative version of Kontsevich's formality ofthe little d-disks operad and of Sinha's cosimplicial model for the space oflong knots arising from Goodwillie-Weiss embedding calculus. As anotheringredient in our proof, we introduce a generalization of a construction thatassociates a cosimplicial object to a multiplicative operad. Along the way wealso establish some results about the Bousfield-Kan spectral sequences of atruncated cosimplicial space.Comment: 25 pages, 2 figure

The rational homology of spaces of long knots in codimension>2