Abstract
In the last decades, Apollonian packings have drawn increasing attention due totheir applications in number theory, geometric group theory, hyperbolic geometry,fractal structures and discrete geometry. In this thesis, we introduce a class of spherepackings where the combinatorics is carried by an edge-scribed polytope. Through thisconnection, we generalize the Apollonian packings in different geometric settings andin higher dimensions. This polytopal structure also allows us to obtain a generalizationof the Descartes’ theorem for the sphere packings which are based on regular polytopes in every dimension. We use the polytopal generalization of the Descartes’ theorem to find characterizations for the integrality of the Apollonian packings based on the Platonic solids. Then, we introduce the notion of Apollonian section, and we use itto show that the set of curvatures of any integral tetrahedral, octahedral or cubicalApollonian packing is contained in the set of curvatures of an integral orthoplicialApollonian packing. The polytopal approach that we propose also allows us to extend the applications of Apollonian packings into a novel direction in the area of topology. In this thesis, we introduce two methods of construction of necklace representations of knots and links. The first method follows directly from the Koebe-Andreev-Thurston Circle Packing Theorem and gives a linear upper bound on the minimum number of spheres needed to construct a necklace representation in terms of the crossing number. In the second method, we use the fractal structure of orthoplicial Apollonian packings to construct necklace representations of rational links with interesting arithmetical properties.