| The goal of this thesis is to relate the projection diagram of a knot or link in S3 to the geometry and topology of the link complement. We use the diagram of a link K to obtain a Dehn surgery description of K from a hyperbolic link L. The simple geometry of S 3L allows us to decompose it into ideal hyperbolic polyhedra, whose dihedral angles provide a lot of combinatorial information. One consequence of this approach is a mild condition on the original diagram that ensures K is hyperbolic and all its non-trivial Dehn fillings are hyperbolike. Another, closely related, consequence is a diagrammatic lower bound on the genus of K.; When K is an arborescent link, we use the correspondence between the link and a weighted tree to simplify the projection diagram into a particularly nice form. This simplified diagram then allows us to subdivide the link complement into hyperbolic polyhedra and tetrahedra whose dihedral angles fit together in a consistent fashion. An angled decomposition of this type implies that K is hyperbolic and provides a robust combinatorial framework for more detailed investigations into its geometry. |
