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Submanifolds in contact geometry

John B. Etnyre

TL;DR

Submanifolds play a central role in the development of contact geometry. The paper surveys how Legendrian and transverse knots drive construction and classification of contact structures in dimension 3, including the tight/overtwisted dichotomy via the Bennequin bound and modern knot invariants, and how knot theory aids in distinguishing and classifying tight structures. It then extends the discussion to higher dimensions, detailing isotropic/coisotropic surgeries, the h-principle for codimension‑2 embeddings, and modern invariants (DGA and knot contact homology) for distinguishing Legendrian submanifolds and contact submanifolds. Together, these perspectives reveal a deep and productive interplay between submanifold theory and global contact topology, with many open questions and active directions for future research.

Abstract

We survey what is known about various special types of submanifolds of contact manifolds and discuss their role in the development of contact geometry.

Submanifolds in contact geometry

TL;DR

Submanifolds play a central role in the development of contact geometry. The paper surveys how Legendrian and transverse knots drive construction and classification of contact structures in dimension 3, including the tight/overtwisted dichotomy via the Bennequin bound and modern knot invariants, and how knot theory aids in distinguishing and classifying tight structures. It then extends the discussion to higher dimensions, detailing isotropic/coisotropic surgeries, the h-principle for codimension‑2 embeddings, and modern invariants (DGA and knot contact homology) for distinguishing Legendrian submanifolds and contact submanifolds. Together, these perspectives reveal a deep and productive interplay between submanifold theory and global contact topology, with many open questions and active directions for future research.

Abstract

We survey what is known about various special types of submanifolds of contact manifolds and discuss their role in the development of contact geometry.

Paper Structure

This paper contains 19 sections, 6 theorems, 8 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Building contact structures
  3. Understanding properties of contact structures
  4. Distinguishing tight contact structures
  5. The structure of $1$-dimensional submanifolds in contact $3$-manifolds
  6. Classification results
  7. Knots in $(S^3,\xi_{std})$
  8. Legendrian links in $(S^3,\xi_{std})$
  9. Legendrian links in other tight contact manifolds
  10. The structure of Legendrian knots
  11. Knots in overtwisted contact structures
  12. Constructing and classifying contact structures via Legendrian knots
  13. Submanifolds in higher dimensions
  14. Constructing contact structures
  15. Distinguishing Legendrian submanifolds
  16. ...and 4 more sections

Key Result

Theorem 2.1

If $\eta$ is a homotopy class of plane field on a compact, oriented manifold $M$, then there is a contact structure $\xi$ in the homtopy class $\eta$.

Figures (5)

  • Figure 1: On the left is the mountain range for the unknot (when $t=-1$), the figure eight knot (when $t=-3$), and the positive twist knots (when $t=-m-1$ for an even number $m$ of twists). Positive twist knots with an odd number $m$ of twists are shown in the middle. The right diagram is an example of the mountain range for a negative $(p,q)$-torus knots; the number of peaks and depth of the valleys depend on $p$ and $q$, and the peaks are at a height of $pq$. The diagonal lines indicate stabilizations.
  • Figure 2: On the left is the mountain range for the negative twist knots. When there are $m$ twists and $m$ is odd, $t=-3, a=-(m+1)/2,$ and $b=1$, while when $m$ is even, $t=1, a=\lceil m^2/8\rceil,$ and $b=-\lceil m/2\rceil$. On the right is the mountain range for the $(3,2)$-cable of the right-handed trefoil.
  • Figure 3: On the left, we see the mountain range for the non-loose unknots in $S^3$ where the bottom vertex is at $(0,1)$. All three figures can be the mountain range for a non-loose rational unknot in some lens space. The coordinates of the bottom vertex and whether one has a "V" as "forward slash" or "back slash" depend on the lens space and contact structure.
  • Figure 4: Possible mountain ranges for non-loose Legendrian torus knots in $S^3$.
  • Figure 5: The projections from $\mathbb{R}^5$ to $\mathbb{R}^3$ of three Legendrian spheres distinguished by the DGA. On the left, the tube about the red path is added along the curved path next to it, and the bottom part of the picture is pushed along the arrow to "stabilize" the Legendrian sphere.

Theorems & Definitions (6)

  • Theorem 2.1: Martinet 1971, Martinet71 and Lutz 1977, Lutz77
  • Theorem 2.2: Ding and Geiges 2004, DingGeiges04
  • Theorem 2.3: Conway 2019, Conway19
  • Theorem 3.1: Bennequin 1983, Bennequin83
  • Theorem 4.1: Etnyre and Van Horn-Morris 2011, EtnyreVanHorn-Morris11
  • Theorem 6.1: Conway and Etnyre 2020, ConwayEtnyre20caps