Plabic Tangles and Cluster Promotion Maps

TL;DR

The work develops a comprehensive framework linking plabic graphs, vector-relation configurations, and promotion maps to the geometry of the amplituhedron and the cluster structure of Grassmannians. It introduces plabic tangles and a dominant solvable criterion, proving several promotion maps (star, spurion, chain-tree, forest) are quasi-cluster homomorphisms and revealing an operadic structure governing promotions. A central tool is the m-intersection number IN_m(G), which equals the count of m-VRCs with fixed boundary and governs when promotion maps invert the amplituhedron map on positroids, with amplitrees achieving IN_m(G)=1. The paper also extends beyond the cluster framework by analyzing the 4-mass box promotion (IN_m(G)=2), exposing positivity properties and suggesting a broader algebraic structure for amplituhedron tiles and Yangian invariants.

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 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.

Figures (13)

• Figure 1: Top: reading left to right, a black-expand move and its effect on $m$-VRCs. Reading right to left, a black-contract move and its effect on $m$-VRCs. Bottom: white-expand and white-contract moves and their effect on $m$-VRCs.
• Figure 2: A square move and its effect on $m$-VRCs when $ce-af \neq 0$. Note that if $ce-af =0$, there is no relation among $u, y, z$ (resp. $v, y, z$) unless $u=0$ (resp. $v=0$).
• Figure 3: A plabic tangle with two blobs; the core is shown in black, while the blobs (inner disks) with their attaching segments are shown in red.
• Figure 4: A brushing for the tangle in \ref{['fig:tangle']}. On the left, the reverse acyclic perfect orientation and paths for $D^{(1)}$, and on the right, those for $D^{(2)}$. The paths are highlighted in blue. Note that some paths in the brushing are trivial and consist of a single boundary vertex.
• Figure 5: Brushed tangles for unary star promotion for $m=3$ and $m=5$ . The labels of the inner disk vertices indicate the brushing: the paths go from the boundary vertex $i \in B_{bd}$ to the (black vertex attached to the) inner disk vertex labeled $i$. These paths extend to a reverse perfect orientation.

Theorems & Definitions (215)

• Conjecture : \ref{['conj:cluster1']}
• Theorem
• Theorem : \ref{['prop:tree1']}
• Theorem : \ref{['th:4mb_pos']}
• Definition 2.1
• Definition 2.2
• Definition 2.3: Positive Grassmannian and positroids
• Lemma 2.4
• proof
• Definition 2.5: Quiver Mutation