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

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 realizes the Feynman diagrams appearing in low-dimensional and quantum topology. Concretely, $L(m)$ is filtered by Cₙ-equivalence, for n≥ 1, which is defined via local moves that can be seen as higher crossing changes. The graded object associated to the Goussarov-Habiro filtration is the Goussarov-Habiro Lie algebra of string links L L(m). We give a concrete presentation, in terms of primitive Feynman (tree) diagrams and relations (1T, AS, IHX, STU²), of the rational Goussarov-Habiro Lie algebra L L(m)Q. 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 Cₙ-equivalence.