Plabic Tangles and Cluster Promotion Maps
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912918861250560 |
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| author | Even-Zohar, Chaim Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren |
| author_facet | Even-Zohar, Chaim Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren |
| contents | Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Plabic Tangles and Cluster Promotion Maps Even-Zohar, Chaim Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 13F60, 05E14, 14M15, 81T99 Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. |
| title | Plabic Tangles and Cluster Promotion Maps |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 13F60, 05E14, 14M15, 81T99 |
| url | https://arxiv.org/abs/2508.02891 |