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Abstract
Let L be a link in S('3) and that S('3)-L is hyperbolic. A method is described which leads to a system of n complex equations in n unknowns such that a solution to the system will yield the explicit hyperbolic structure of S('3)-L in terms of a set of ideal hyperbolic tetrahedra with identifications on pairs of faces. This allows one to calculate such useful invariants as the hyperbolic volume. A computer implementation of this method is given along with numerous examples and tables of results.
Next, it is shown that a certain extension of the class of prime alternating non 2-braid links all have hyperbolic complements. This is used to show that many sequences of hyperbolic link complements converge to other hyperbolic link complements and then that all closed 3-manifolds come from surgery on hyperbolic link complements.
We then prove that thrice-punctured spheres in finite volume hyperbolic 3-manifolds are isotopic to totally geodesic thrice-punctured spheres. This is used to show that many nonhomeomorphic link complements have the same volume and then that certain types of links when constructed from simpler hyperbolic links, will also be hyperbolic with volume the sum of the volumes of the simpler links.
Finally, using all of the previous results, we derive bounds on the volumes of certain types of hyperbolic link complements.
