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

TL;DR

This paper provides a concrete presentation of the rational Goussarov-Habiro Lie algebra of string links using primitive Feynman diagrams and relations, connecting topology, graph theory, and quantum invariants.

Contribution

It introduces a new explicit diagrammatic presentation of the rational Goussarov-Habiro Lie algebra for string links, utilizing primitive Feynman diagrams and graph cycle analysis.

Findings

Presented a concrete diagrammatic description of the Lie algebra.

Connected graph cycles and STU relations to the algebra's structure.

Provided an alternative proof of Massuyeau's rational Goussarov-Habiro conjecture.

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 realizes 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 crossing changes. The graded object associated to the Goussarov-Habiro filtration is the Goussarov-Habiro Lie algebra of string links $\mathcal{L} L(m)$. We give a concrete presentation, in terms of primitive Feynman (tree) diagrams and relations ($\text{1T}$, $\text{AS}$, $\text{IHX}$, $\text{STU}^2$), of the rational Goussarov-Habiro Lie algebra $\mathcal{L} L(m)_{\mathbb{Q}}$. To that end, we investigate cycles in graphs of forests: flip graphs associated to forest diagrams and their $\text{STU}$ 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.