The slack model in the study of polytopes (2020)
In this thesis we present some of the applications of studying polytopes via their slack matrices and slack ideals. We prove that McMullen's operations on polytopes, which includes the join, the vertex sum, the vertex splitting and their dual operations, preserve graphicality in the same way it was known it preserves projective uniqueness. We use this result to identify a large class of projectively unique order polytopes; namely, we prove that every ranked finite poset with no 3-antichain has a graphic and thus projectively unique order polytope. We give a complete characterization of complex psd-minimal polygons; we prove that the complex psd-minimal polygons are precisely the triangles, the quadrialterals and a special class of hexagons which we call Pappus' hexagons. Using this, it can be seen that complex psd-minimal 3-polytopes must have vertices of degree 3, 4 or 6 and facets with 3, 4 or 6 sides. We identify all the combinatorial classes of 3-polytopes...