Legendrian Hopf links in L(p,1)

TL;DR

This work classifies Legendrian realisations of the positive Hopf link in lens spaces $L(p,1)$ across all contact structures, extending the known S^3 results. It develops a framework based on rational invariants $ t tb_{b Q}$, $ t rot_{b Q}$, and the $d_3$-invariant, and computes these via surgery diagrams and contact cuts, enabling a finite, case-by-case classification. The main finding is that in the tight complement, the Hopf link is Legendrian simple, with a precise enumeration of exceptional realizations and explicit constructions, including a novel contact-cut approach for case (c1). The results provide the first comprehensive Legendrian link classification in a 3-manifold other than $S^3$, illustrating how contact cuts and rational invariants interplay to distinguish loose versus exceptional components.

Abstract

We classify Legendrian realisations, up to coarse equivalence, of the Hopf link in the lens spaces L(p,1) with any contact structure.

Figures (15)

• Figure 1: The positive Hopf link in $L(p,1)$.
• Figure 2: Legendrian Hopf links in $(L(p,1),\xi_{\mathrm{tight}})$. Here $k+\ell=p$.
• Figure 3: Kirby moves for (c3).
• Figure 4: Kirby moves for (d1).
• Figure 5: Kirby moves for (d2).

Theorems & Definitions (13)

• Lemma 1.1
• proof
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Remark 4.1
• Lemma 5.1
• proof
• Remark 5.2