Hyperbolic Structures from Link Diagrams

As a result of Thurston's Hyperbolization Theorem, many 3-manifolds have a hyperbolic metric or can be decomposed into pieces with hyperbolic metric (W. Thurston, 1978). In particular, Thurston demonstrated that every link in a 3-sphere is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating link represented by a prime diagram is either hyperbolic or a (2,n)-torus link.

A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on ideal polygons bounding the regions of a diagram of the link rather than decomposition of the complement into ideal tetrahedra, was suggested by M. Thistlethwaite. The method allows one to compute the geometric structure directly from the 2-dimensional projection of the link, together with overcrossing-undercrossing data. It is applicable to all diagrams of hyperbolic links under a few mild restrictions.

The author introduces the basics of the method, as well as its consequences. In particular, a surprising rigidity property of certain tangles is discussed, as well as its applications. A new numerical invariant for tangles is introduced, and a field that is a topological invariant for hyperbolic links. It is shown how to calculate the holonomy representation of link group using the new method, and how the labels coming from the method reflect the geomtry of the polyhedral decomposition of a link complement. Also, formulae that allow one to calculate the exact hyperbolic volume, as well as complex volume, of 2-bridged links directly from their diagrams, are obtained.