Here are some conjectures which should not be difficult to prove or disprove about the 1-skeleton of these polytopes.

• Its an infinite tree (i.e. acyclic) allowing that some edges (I’ll call them “leaves”)will go to infinity and therefore have only one vertex.
• For generic polytopes every vertex has order three.
• Each edge divides the tree into finite and infinite parts, thus giving a natural orientation for each edge pointing toward the infinite part.
• With the edges so oriented every vertex will have two incoming and on outgoing edge, and there will be a bijection between cells and vertices given as: the edges of a particular cell all flow toward one of its vertices and for that vertex its two incoming edges both border on that cell.
• Starting from any edge traversing around the finite trees finite side back to that edge will take you through consecutively numbered cells (numbering the cells, as in the previous post, by the portion of the root they are perpendicular to)

A lot of material, some very scattered and some a little organized, should be written up. Very old material even, before my orals. If ever I can’t find something to do with myself I can always try to write up summaries of what I know about

1. Sheaves,
2. Derived things,
3. Intersection Homology, and
4. newer stuff…