Polytopes with mass linear functions II: The four-dimensional case

This paper continues the analysis begun in Polytopes with mass linear functions, Part I of the structure of smooth moment polytopes that support a mass linear function. As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism π₁(Tⁿ)→π₁(Symp(M_Δ, ω_Δ)) from the fundamental group of the torus T ⁿ to that of the group of symplectomorphisms of the 2n-dimensional symplectic toric manifold (M_Δ,ω_Δ) associated to Δ. In Part I, we made a general investigation of this question and classified all mass linear pairs (Δ,H) in dimensions up to 3. The main result of the current paper is a classification of all four-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blow ups, and expansions (or wedges), describing their effect on both moment polytopes and mass linear functions. We end by discussing the relation of mass linearity to Shelukhin's notion of full mass linearity. The two concepts agree in dimensions up to and including 4. However, full mass linearity may be the more natural concept when considering the question of which blowups preserve mass linearity.