Generalization of the Thistlethwaite--Tsvietkova Method

TL;DR

The paper advances a general framework that extends the Thistlethwaite--Tsvietkova method from link complements to cusped hyperbolic 3-manifolds via ideal polyhedral decompositions, tying algebraic labelings to ${PSL(2,\mathbb{C})}$-representations and showing the maximal-volume solution corresponds to the complete hyperbolic structure.It introduces a geometric realization that translates algebraic data into a parabolic representation, enabling a volume-maximization principle and a direct link between labels, representations, and hyperbolic geometry.The approach specializes gracefully to fully augmented links, links in thickened tori, and related circle-packings; it establishes a precise 1-1 correspondence between algebraic solutions (under certain criteria) and circle packings realizing the region graph, with additional criteria yielding univalence and local order-preservation.Together with explicit equation systems for FALs in $S^3$ and in $T^2\times(-1,1)$ and illustrative examples, the work provides a practical, diagram-driven toolkit for recovering complete hyperbolic structures from combinatorial data and enriching the interaction between 3-manifold topology, hyperbolic geometry, and circle packing theory.

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.

Figures (14)

• Figure 1: (a) Link diagram of $K$ (b) crossing circles added to each twist region (c) the third picture is a fully augmented link diagram with all full-twists removed (d) fully augmented link diagram with no half-twists. Note that removing a full-twist from a FAL does not change the homeomorphism class of its complement, but removing a half-twist might.
• Figure 2: A brief review of the cut-slice-flatten method of lackenby, kwon2020 for a FAL with no half-twists: (a) A fundamental domain for a fully augmented square weave, $L$. (b) Disks cut in half at each augmentation circle. (c) Sliced and flattened half-disks at each augmentation circle (d) Collapsing the bold strands in (c) to ideal points gives the bow-tie graph $B_{L}$. The half-disks become bow-ties, i.e. a shaded pair of triangular regions; the white regions are the $F_R$'s corresponding to regions of the link diagram.
• Figure 3: Another example of a FAL diagram $D_L$ and corresponding bow-tie graph $B_L$; the arrows on the vertices in $B_L$ indicate the orientations of their corresponding components/segments of $L$; for $C_i$, the arrow follows the direction of the top half of $C_i$.
• Figure 4: Notation for crossing arcs and peripheral edges near spanning face without (a) and with (b) half-twist. By default, we orient $e_i⁰$ is oriented following $C_i$, $e_{i,±}¹,e_{i,±}²$ are oriented by right-hand thumb rule around $L$, and $e_i^{μ1},e_i^{μ2}$ are oriented by the right-hand thumb rule around $C_i$. The order between $γ_i¹,γ_i²$ is chosen so that $e_{i,+}⁰$ is oriented from $γ_i¹$ to $γ_i²$. For the spanning disk region equations (\ref{['e:spanning-disk-eqn']}), $χ_i¹ = +1$ and $χ_i² = -1$. (c) Crossing arcs and peripheral edges depicted in $B_L$, for $L$ with no half-twist.
• Figure 5: Diagrams modified from kwontham depicting a (non-continuous) operation on a FAL complement: at an augmentation circle with a half-twist, slice along the spanning twice-punctured disk, rotate one side by $180^{∘}$ to undo the half-twist, then glue the two sides back.

Theorems & Definitions (110)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Remark 2.9
• Definition 2.10