Séminaire Lotharingien de Combinatoire, 95B.62 (2026), 12 pp.
Chaim Even-Zohar, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler and Lauren Williams
Plabic Tangles and Cluster Promotion Maps
Abstract.
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. 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,
providing some generating functions.
Promotion maps have important connections to the geometry and cluster
structure of the amplituhedron and singularities of scattering
amplitudes in planar N = 4 super Yang-Mills theory.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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