On the complexity of 2-bridge link complements

TL;DR

The paper addresses the complexity of hyperbolic 2-bridge link complements by leveraging Sakuma–Weeks triangulations and angle-structure techniques to link hyperbolic volume to minimal triangulations. It re-establishes a fundamental non-minimality condition for these triangulations and, for links meeting the condition, constructs explicit angle structures to obtain lower bounds on complexity from volume, yielding multiplicative and additive bounds that improve on prior estimates. The analysis introduces a block-decomposition framework for the 2-bridge words, assigns shapes to tetrahedra to maximize volume within gluing constraints, and derives bounds such as c(M)  0.8|T| with refined additive bounds expressed in terms of syllable structure, while also comparing to existing volume-based bounds. Overall, the work clarifies when Sakuma–Weeks triangulations are minimal and provides practical, sharp lower bounds on the complexity of an infinite family of 2-bridge link complements. The results have implications for computational topology and the understanding of how combinatorial triangulations encode hyperbolic geometry in 3-manifolds.

Abstract

We reprove a necessary condition for the Sakuma-Weeks triangulation of a 2-bridge link complement to be minimal in terms of the mapping class describing its alternating 4-string braid construction. For the 2-bridge links satisfying this condition we construct explicit angle structures on the Sakuma-Weeks triangulations and compute both multiplicative and additive lower bounds on the complexity of the link complements via volume estimates. These lower bounds are an improvement on existing volume estimates for the 2-bridge links examined.

Figures (8)

• Figure 1: Crossings of the four arcs encoded by $R$ and $L$. A crossing encoded by $R$ is called vertical and a crossing encoded by $L$ is called horizontal.
• Figure 2: The $2$-bridge link $K(\Omega)$ with $\Omega = R^3L^2R$. Removing the outermost and innermost crossings returns the four-string braid corresponding to $\Omega$.
• Figure 3: Ideal triangulation of $S_i\times\{1\}$ (outside) and $S_i\times\{0\}$ (inside) with orientations marked on each pair of edges. The complements of the four arcs define an isotopy between the ideal triangulations. Going from $S_i\times\{1\}$ to $S_i\times\{0\}$, a vertical twist $R$ exchanges vertical and diagonal edges and a horizontal twist $L$ exchanges horizontal and diagonal edges, as indicated by the braid arcs (dashed lines).
• Figure 4: The process of gluing the layers of the Sakuma-Weeks triangulation together. Each layer consists of two ideal tetrahedra glued along their horizontal and vertical edges. The four faces on the back of $\widetilde{\Delta_i}$ lie on $S_i\times\{0\}$ and the four faces on the front lie on $S_{i+1}\times\{1\}$ and similarly for $\widetilde{\Delta_{i+1}}$. We have highlighted two faces on $S_{i+1}\times\{1\}$ and the faces they glue to in $S_{i+1}\times\{0\}$.
• Figure 5: Identifications of remaining ideal triangles in $\widetilde{\Delta}_1$ and $\widetilde{\Delta}_2$. For each layer shown, the back faces belong to $S_i\times\{0\}$ and the front faces belong to $S_{i}\times\{1\}$. Note that no two faces of a single ideal tetrahedron are identified.

Theorems & Definitions (42)

• Conjecture 1
• Theorem 2: menasco_closed_1984, Corollary 2
• Definition 3
• Theorem 4
• Definition 5
• Theorem 6
• Corollary 7
• Lemma 8
• proof
• Lemma 9