Generalization of the Thistlethwaite--Tsvietkova Method

Abstract

Thurston's equations determine the hyperbolic structure of a 3-manifold with a triangulation. In work by Thistlethwaite and Tsvietkova, an alternative method was developed for link complements in $S^3$ depending on the link diagram, where a set of labels are associated to the vertices and edges of the link diagram, and one attempts to solve a set of equations on the labels. Under certain conditions, there exists a solution to these equations that corresponds to the complete hyperbolic structure, but in general it is difficult to determine which one it is. We generalize this method to 3-manifolds with a polyhedral decomposition, and show that solutions to the equations correspond to $PSL(2,\mathbb{C})$-representations of the fundamental group, and that the solution with the largest volume corresponds to the complete hyperbolic structure. We also consider different classes of complements of links, in particular links in the thickened torus and fully augmented links. For the latter, we establish a correspondence between solutions satisfying some criteria and circle packings realizing the region graph associated to the fully augmented link.