Hodge‐type decomposition in the homology of long knots

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Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011, methodology): “It is well-known that the space Embc(R,R), n > 2m+2, is a connected (m + 1)-loop space [6, 40].”
The rational homotopy of mapping spaces of E${}_n$ operads (2017, methodology): “We refer to the articles [1, 2, 39, 47], where the hairy graph complexes is defined and studied, for more details on our claims.”
On the rational homology of high dimensional analogues of spaces of long knots (2011, result): “In the case mD 1, this double splitting is equivalent to the one studied by the second author in [23].”
M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra (2010, background): “de aware by V. Turchin recently that the deformation complexes of operad maps em→en, but with the bracket being sent to zero, have also been considered and related to a version of graph cohomology in [43, 5, 4, 29]. 1.1. Structure of the paper. We tried to keep the main body of this paper short, at ”