Primitive Feynman diagrams and the rational Goussarov–Habiro Lie algebra of string links

Primitive Feynman diagrams and the rational Goussarov-Habiro Lie algebra of string links BRUNO DULAR Goussarov-Habiro's theory of clasper surgeries defines a filtration of the monoid of string links L.m/ on m strands, in a way that geometrically realises the Feynman diagrams appearing in low-dimensional and quantum topology.Concretely, L.m/ is filtered by C n -equivalence, for n 1, which is defined via local moves that can be seen as higher-order crossing changes.The graded object associated to the Goussarov-Habiro filtration is the Goussarov-Habiro Lie algebra of string links LL.m/.We give a concrete presentation, in terms of primitive Feynman (tree) diagrams and relations (1T, AS, IHX, STU 2 ), of the rational Goussarov-Habiro Lie algebra LL.m/ Q and of the primitive Lie algebra of the Hopf algebra of Feynman diagrams.To that end, we investigate cycles in graphs of forests: flip graphs associated to forest diagrams and their STU relations.As an application, we give an alternative diagrammatic proof of Massuyeau's rational version of the Goussarov-Habiro conjecture for string links, which relates indistinguishability under finite type invariants of degree < n and C n -equivalence.