Tops in Higher Dimensions
Reflexive polytopes can be used to construct families of Calabi-Yau varieties as hypersurfaces in toric varieties. Reflexive polytopes have been classified for $n \leq 4$ by constructing maximal objects that contain all reflexive polytopes as subsets. Given computational constraints, classification becomes intractable for $n = 5$. An alternate approach is to make use of tops, which are generalizations of the notion of half of a reflexive polytope. In this thesis, we investigate classification algorithms for tops and construct a class of four-dimensional tops which can be completed to reflexive polytopes. We then outline how this all relates to fiber bundles on toric varieties.