Non-loose Legendrian Hopf links in lens spaces

TL;DR

This work resolves the full coarse-classification of non-loose Legendrian Hopf links in lens spaces $L(p,q)$, extending earlier results to include cases with nonzero Giroux torsion. The authors develop a framework combining convex surface theory and Farey-graph technology to enumerate possible boundary-slopes decompositions, compute rational invariants $\text{tb}_{\mathbb{Q}}$ and $\text{rot}_{\mathbb{Q}}$, and track how stabilizations relate distinct non-loose realizations. They provide explicit classifications for $L(p,1)$ and $L(2n+1,2)$, describe their mountain-range structures, and extend the classification to general $L(p,q)$, detailing cases with zero torsion and those with Giroux torsion in the complement. An algorithmic connection to contact surgery diagrams is developed, enabling the construction of explicit surgeries for the non-loose Hopf links, thereby linking convex-geometric data to concrete Kirby moves. The results advance understanding of non-loose links in 3-manifolds and furnish practical tools for constructing tight structures via surgery in lens spaces.

Abstract

We give a complete classification of non-loose Legendrian Hopf links in $L(p,q)$ generalizing a result of the author with Geiges and Onaran. The classification is for non-loose Hopf links for both zero and non-zero Giroux torsion in their complement. We also give an explicit algorithm for the contact surgery diagrams for all these Legendrian representatives with no Giroux torsion in their complement.

Figures (20)

• Figure 1: Hopf link in $L(p,q)$ where $-p/q=[a_0,a_1\cdots, a_n]$
• Figure 2: (a) A backward slash and a forward slash based at $(a,b)$. (b) On the right we see a $V$ based at $(a,b)$.
• Figure 3: Part of a loose cone with two peaks.
• Figure 4: The loose mountain range of $L_1$ when we fix $L_2$ for $-p/q=[a_0,a_1].$
• Figure 5: The Farey graph.

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• Theorem 2.2