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Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

Dahyana Farias, Eduardo Fernández, Francisco Presas, Guillermo Sánchez-Arellano

TL;DR

The paper addresses the classification problem for convex disks with fixed characteristic foliations and Legendrian boundary in closed overtwisted contact 3-manifolds. It develops an 1-parameter h-principle: under tight foliation or tb(∂D^2)+|rot(∂D^2)|>-1, the inclusion from actual embeddings to formal F-embeddings is an isomorphism on path components with trivial relative π1, aligning geometric and formal data. The authors then derive substantial consequences for spaces of Legendrian unknots in Darboux balls, compute the fundamental group of Legendrian unknot spaces in overtwisted S^3 (notably π1 ≅ ℤ⊕ℤ in S^3), and describe the contact mapping class group of complements of such unknots, including exotic ℤ2 contributions in certain OT components. The approach combines Eliashberg’s overtwisted h-principle with a refined analysis of isotopies of overtwisted disks, bypass triangles, and discretized isotopies to handle homotopy obstructions, yielding rigorous classifications and new insights into the topology of Legendrian submanifolds in OT settings.

Abstract

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood of such a disk is tight or where the boundary violates the Bennequin-Eliashberg inequality. We show that this classification coincides with the formal one, establishing an h-principle for these disks. As a corollary, we deduce that the space of Legendrian unknots that lie in some Darboux ball in a closed overtwisted contact 3-manifold satisfies the h-principle at the level of fundamental groups. Finally, we determine the contact mapping class group of the complement of each Legendrian unknot with non-positive tb invariant in an overtwisted 3-sphere.

Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

TL;DR

The paper addresses the classification problem for convex disks with fixed characteristic foliations and Legendrian boundary in closed overtwisted contact 3-manifolds. It develops an 1-parameter h-principle: under tight foliation or tb(∂D^2)+|rot(∂D^2)|>-1, the inclusion from actual embeddings to formal F-embeddings is an isomorphism on path components with trivial relative π1, aligning geometric and formal data. The authors then derive substantial consequences for spaces of Legendrian unknots in Darboux balls, compute the fundamental group of Legendrian unknot spaces in overtwisted S^3 (notably π1 ≅ ℤ⊕ℤ in S^3), and describe the contact mapping class group of complements of such unknots, including exotic ℤ2 contributions in certain OT components. The approach combines Eliashberg’s overtwisted h-principle with a refined analysis of isotopies of overtwisted disks, bypass triangles, and discretized isotopies to handle homotopy obstructions, yielding rigorous classifications and new insights into the topology of Legendrian submanifolds in OT settings.

Abstract

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood of such a disk is tight or where the boundary violates the Bennequin-Eliashberg inequality. We show that this classification coincides with the formal one, establishing an h-principle for these disks. As a corollary, we deduce that the space of Legendrian unknots that lie in some Darboux ball in a closed overtwisted contact 3-manifold satisfies the h-principle at the level of fundamental groups. Finally, we determine the contact mapping class group of the complement of each Legendrian unknot with non-positive tb invariant in an overtwisted 3-sphere.
Paper Structure (27 sections, 31 theorems, 42 equations, 8 figures)

This paper contains 27 sections, 31 theorems, 42 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Legendrian unknots
  4. The contact mapping class group of a Legendrian unknot with negative $\operatorname{tb}$ in an overtwisted $3$-sphere
  5. Further questions
  6. Outline
  7. Preliminaries
  8. Convex Surface Theory
  9. Invariants of Legendrian knots
  10. Bypass basics
  11. The formal class of a bypass triangle
  12. Formal ${\mathcal{F}}$-embeddings
  13. Overtwisted $h$-principles
  14. Overtwisted $h$-principle
  15. Loose Overtwisted $h$-principle
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $(M,\xi)$ be a closed overtwisted contact $3$-manifold and $\mathbb{D}^2\subseteq (M,\xi)$ an embedded convex disk with Legendrian boundary and characteristic foliation $\mathcal{F}$. Assume that either ${\mathcal{F}}$ is tight or $\operatorname{tb}(\partial\mathbb{D}^2)+|\operatorname{rot}(\par induces an isomorphism at the level of path-connected components. Moreover, the first relative homo

Figures (8)

  • Figure 1: (a). Schematic of the two disks $e_0$ and $e_1$. The situation is described in the solid torus complementary to the boundary of the disks. The pinched $3$-disk $(D,\xi)$ is depicted in gray, the tight region is depicted in green. (b) Effect of applying the contactomorphism $\varphi$ to the pinched $3$-disk $(D,\xi)$.
  • Figure 2: A positive bypass, where we can see in light blue the attaching arc, in black the Honda arc, in red the dividing set and in gray the characteristic foliation.
  • Figure 3: Result of attaching a bypass from above. The effect of attaching a bypass from below is the mirror of this figure.
  • Figure 4: Bypass triangle
  • Figure 5: Overtwisted disk. The dividing set is depicted in red and the characteristic foliation in gray.
  • ...and 3 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Corollary 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 47 more