Plabic Tangles and Cluster Promotion Maps

TL;DR

This paper introduces plabic tangles and promotion maps, connecting plabic graphs to Grassmannian transformations, with implications for the geometry of the amplituhedron and scattering amplitudes.

Contribution

It defines promotion maps via $m$-VRCs, proves they are quasi-cluster homomorphisms in certain cases, and explores their operad structure and positivity properties.

Findings

Promotion maps relate to amplituhedron geometry and cluster structures.

Proved promotion maps are quasi-cluster homomorphisms for several classes.

Identified positivity properties for non-rational maps beyond cluster algebras.

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.