Geometry of fully augmented links in doubled 3-manifolds

TL;DR

This work generalizes the hyperbolic geometry of classical fully augmented links to fully augmented links embedded on reflection surfaces within doubles of compact 3-manifolds, including virtual variants. By defining FALs on a surface, leveraging reflective symmetry from doubling, and applying circle-packings and rigidity results, the authors obtain concrete cusp-shape bounds and volume estimates. They establish lower bounds on volume via Miyamoto’s theorem and derive explicit upper bounds for virtual links, with sharp results in genus 1, and extend Dehn-filling techniques to obtain volume-change bounds. The results broaden the applicability of fully augmented link geometry to new 3-manifold settings and connect to virtual/shadow link theory and Dehn-filling phenomena.

Abstract

Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by doubling a compact 3-manifold and show that the results of the classical setting extend to these links. When the resulting manifolds are hyperbolic, we find bounds on their cusp shapes and volumes. Note these links include virtual fully augmented links, and thus our bounds apply to such links when they are hyperbolic.

Figures (6)

• Figure 2.1: Cutting and regluing at a crossing circle to add a half-twist.
• Figure 3.1: Rectangle with circle white sides
• Figure 4.1: How the tiling of horospherical cusp tori changes when adding a half-twist at a crossing circle.
• Figure 4.2: If $H$ is centred at a point $p$ on a white face $W$, and $H$ has diameter greater than one, then it must contain the midpoint of an edge through $p$. This is Figure 7.17 of Purcell:HyperbolicKnotTheory
• Figure 6.1: A hyperideal tetrahedron in ${\mathbb{H}}^3$. The hyperideal vertex may be viewed as lying beyond $\partial_\infty {\mathbb{H}}^3$.

Theorems & Definitions (43)

• Theorem 1.1: Cusp shapes of fully augmented links
• Theorem 1.2: Lower volume bounds
• Theorem 1.3: Upper volume bounds, virtual setting
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Definition 2.5
• Example 2.6