Given an algebraic hypersurface H in (C^*)^n, I will define several polyhedral objects and show that they are homeomorphic to H. The first one is “phase tropical”, which is a certain degeneration limit of H which is a fibration over the tropical hypersurface with fibers given by coamoebas of the corresponding local polynomials. It was conjectured by Viro that it is a topological manifold. The second one is “ober-tropical”, where the coamoeaba fibers are replaced by their (permutahedral) skeleta. The last one is when the permutahedral skeleta are replaced by “soft cells”. This one has a smooth structure induced from the ambient space, and, in fact, can be shown to be symplectomorphic to the original hypersurface where the fibers are Lagrangian skeleta. I will concentrate on the case of the pair-of-pants.

Based on several past works with G. Kerr and H. Ruddat and on some ongoing projects with C.-Y. Mak, D. Matessi, H. Ruddat, and A. Vicente.