Generalization of the Thistlethwaite--Tsvietkova Method

TL;DR

This paper extends the Thistlethwaite--Tsvietkova method to a broader class of 3-manifolds, linking solutions of algebraic equations to hyperbolic structures and circle packings, thus broadening its applicability.

Contribution

It generalizes the original method to 3-manifolds with polyhedral decompositions and relates solutions to hyperbolic structures and circle packings.

Findings

Solutions correspond to PSL(2,C)-representations of the fundamental group.

The largest volume solution yields the complete hyperbolic structure.

Established a link between solutions and circle packings for fully augmented links.

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.