Two graph homologies and the space of long embeddings

Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings. Two graph homologies have been invented from two approaches to study the space of long embeddings: the hairy graph homology from embedding calculus, and BCR graph homology from configuration space integral. In this paper, we construct a monomorphism from the top hairy graph homology to the top BCR graph homology, though the latter graph homology is quite modified. This map and its left inverse are analogs of PBW isomorphism between the space of open Jacobi diagrams and the space of Jacobi diagrams on the unit circle, in the theory of Vassiliev knot invariants.