Construction and Analysis of Projected Deformed Products

TL;DR

This paper introduces a new method for constructing deformed products of polytopes that preserve certain faces under projection, enabling the creation of diverse neighborly cubical polytopes with explicit combinatorial descriptions.

Contribution

It develops a novel deformed product construction using matrix representations and Gale duality, allowing explicit analysis and generation of neighborly cubical polytopes and DPPs with preserved face structures.

Findings

Constructed deformed n-cubes with neighborly properties via projection.

Provided explicit combinatorial descriptions of projected polytopes.

Generated a variety of neighborly cubical polytopes and DPPs with high fatness.

Abstract

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.