#### Year of Publication

2016

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Carl W. Lee

#### Abstract

We may describe a polytope *P* as the convex hull of *n* points in space. Here we consider the numbers of chains of faces of *P*. The toric *g*-vector and *CD*-index of *P* are useful invariants for encoding this information. For a simplicial polytope *P*, Lee defined the winding number *w _{k}* in a Gale diagram corresponding to

*P*. He showed that

*w*in the Gale diagram equals

_{k}*g*of the corresponding polytope. In this dissertation, we fully establish how to compute the

_{k}*g*-vector for any polytope with few vertices from its Gale diagram. Further, we extend these results to polytopes with higher dimensional Gale diagrams in certain cases, including the case when all the points are in affinely general position. In the Generalized Lower Bound Conjecture, McMullen and Walkup predicted that if

*g*(

_{k}*P*)=0 for some simplicial polytope

*P*and some

*k*, then

*P*is (

*k*-1)-stacked. Lee and Welzl independently use Gale transforms to prove the GLBC holds for any simplicial polytope with few vertices. In the context of Gale transforms, we will extend this result to nonpyramids with few vertices. We will also prove how to obtain the

*CD*-index of polytopes dual to polytopes with few vertices in several cases. For instance, we show how to compute the

*CD*-index of a polytope from the Gale diagram of its dual polytope when the Gale diagram is 2-dimensional and the origin is captured by a line segment.

#### Digital Object Identifier (DOI)

http://dx.doi.org/10.13023/ETD.2016.080

#### Recommended Citation

Nelson, Sarah A., "FLAG *F*-VECTORS OF POLYTOPES WITH FEW VERTICES" (2016). *Theses and Dissertations--Mathematics*. 33.

https://uknowledge.uky.edu/math_etds/33