Cluster algebras in scattering amplitudes with special 2D kinematics

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We study the cluster algebra of the kinematic configuration space Confₙ(P³) of a n-particle scattering amplitude restricted to the special 2D kinematics. We found that the n-points two loop MHV remainder function found in special 2D kinematics depend on a selection of -coordinates that are part of a special structure of the cluster algebra related to snake triangulations of polygons. This structure forms a necklace of hypercubes beads in the corresponding Stasheff polytope. Furthermore in n = 12, the cluster algebra and the selection of -coordinates in special 2D kinematics replicates the cluster algebra and the selection of -coordinates of n=6 two loop MHV amplitude in 4D kinematics.