Primitive Feynman diagrams and the rational Goussarov--Habiro Lie algebra of string links
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915920771809280 |
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| author | Dular, Bruno |
| author_facet | Dular, Bruno |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive Feynman diagrams and the rational Goussarov--Habiro Lie algebra of string links Dular, Bruno Geometric Topology Quantum Algebra 57K16 (Primary) 16T30, 57M25, 17B70 (Secondary) 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. |
| title | Primitive Feynman diagrams and the rational Goussarov--Habiro Lie algebra of string links |
| topic | Geometric Topology Quantum Algebra 57K16 (Primary) 16T30, 57M25, 17B70 (Secondary) |
| url | https://arxiv.org/abs/2406.01093 |