Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic

TL;DR

The paper extends hyperbolicity results from classical alternating links in $S^3$ to generalized augmented cellular alternating links in thickened orientable surfaces and more general $I$-bundles over closed surfaces. By augmenting reduced cellular alternating projections with nonisotopic trivial components and applying Thurston's hyperbolicity criterion, it shows that the complements admit complete finite-volume hyperbolic metrics, with totally geodesic boundary in key cases; the framework also handles extensions to nonorientable bases via orientable double covers. A new class, rubber band links, is introduced and shown to be hyperbolic, with a construction that preserves hyperbolicity under half-twist operations on thrice-punctured disks, yielding explicit volume bounds in terms of $\chi(S)$ and the number of edges $\varepsilon$ using $v_{oct}$ and $v_{tet}$. The results unify and extend existing volume bounds (Lackenby, KP, Kwon, Adams) to thickened surfaces, culminating in a geometric-topology picture where hyperbolic cellular and augmented cellular links form a closed subset of finite-volume hyperbolic 3-manifolds under Dehn filling and related operations.

Abstract

Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n \geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S \times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S \times I$ to which these results apply are included.

Figures (14)

• Figure 1: Crossings that can be reduced. Here, $T$ refers to the portion of the link projection contained in a disk in the projection surface whose boundary intersects the link exactly twice.
• Figure 2: A generalized augmented cellular alternating link.
• Figure 3: Closing off $E$ with embroidery.
• Figure 4: Suppose the addition of embroidery arcs created a flype crossing in $D'$.
• Figure 5: Complementary regions of embroidery arcs. The central region is shaded and always has the shape of a flower, perhaps with two bisected petals.

Theorems & Definitions (38)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.3
• Definition 1.4
• Definition 2.1
• Theorem 2.1
• Lemma 2.2