Primitive Feynman diagrams and the rational Goussarov--Habiro Lie algebra of string links
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.01093
- arXiv:
- arXiv:2406.01093
- Bibcode:
- 2024arXiv240601093D
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Quantum Algebra;
- 57K16 (Primary) 16T30;
- 57M25;
- 17B70 (Secondary)
- E-Print:
- 42 pages, comments are welcome!