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

Dular, Bruno

doi:10.2140/agt.2026.26.1037

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

Bruno Dular

Algebraic & Geometric Topology 26 (2026) 1037–1076

DOI: 10.2140/agt.2026.26.1037

Abstract

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 $n \geq 1$ , 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 $ℒ L (m)$ . 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 $ℒ L {(m)}_{ℚ}$ 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.

Keywords

knots, string links, finite type invariants, Vassiliev invariants, primitive Feynman diagrams, tree diagrams

Mathematical Subject Classification

Primary: 57K16

Secondary: 16T30, 17B70

References

Publication

Received: 6 August 2024

Revised: 17 March 2025

Accepted: 3 May 2025

Published: 1 April 2026

Authors

Bruno Dular
Department of Mathematics University of Luxembourg Esch-sur-Alzette Luxembourg

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Volume 26, issue 3 (2026)