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Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds

Patricia Cahn, Rima Chatterjee, Vladimir Chernov

TL;DR

The paper addresses the problem of Legendrian simplicity for links in overtwisted 3-manifolds by introducing $V$-transverse (pseudo-Legendrian) links and developing a framework to study simplicity across framed isotopy, $V$-transverse homotopy, and Legendrian link-homotopy. The authors construct infinite families of non-simple isotopy classes in $S^1$-bundles over surfaces endowed with nowhere-vanishing vector fields $V_k$, yielding both $V$-transverse and Legendrian non-simplicity within the same smooth isotopy class. Their approach combines (i) approximating framed isotopies by $V$-transverse isotopies via the kink homomorphism $h_V$, (ii) an explicit generator description of $ ext{π}_1$ of spaces of multicurves in $S^1$-bundles, and (iii) a finite-type invariant $ u$ for $V$-transverse figure-8s that detects obstruction to $V$-transverse link-homotopies. The results reveal that non-simplicity phenomena persist in overtwisted settings and can occur concurrently for $V$-transverse and Legendrian viewpoints, highlighting new flexibility in loose Legendrian theory and providing tools for distinguishing complex link types in contact 3-manifolds.

Abstract

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and have Legendrian-isotopic components and (2) a pair of Legendrian links that are not Legendrian isotopic, but are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and which are link-homotopic as Legendrian links. Moreover, we construct examples showing that both of these non-simplicity phenomena can occur in the same smooth isotopy class. To construct these examples, we develop the theory of links transverse to a nowhere-zero vector field in a 3-manifold, and construct analogous examples in the category of links transverse to a vector field.

Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds

TL;DR

The paper addresses the problem of Legendrian simplicity for links in overtwisted 3-manifolds by introducing -transverse (pseudo-Legendrian) links and developing a framework to study simplicity across framed isotopy, -transverse homotopy, and Legendrian link-homotopy. The authors construct infinite families of non-simple isotopy classes in -bundles over surfaces endowed with nowhere-vanishing vector fields , yielding both -transverse and Legendrian non-simplicity within the same smooth isotopy class. Their approach combines (i) approximating framed isotopies by -transverse isotopies via the kink homomorphism , (ii) an explicit generator description of of spaces of multicurves in -bundles, and (iii) a finite-type invariant for -transverse figure-8s that detects obstruction to -transverse link-homotopies. The results reveal that non-simplicity phenomena persist in overtwisted settings and can occur concurrently for -transverse and Legendrian viewpoints, highlighting new flexibility in loose Legendrian theory and providing tools for distinguishing complex link types in contact 3-manifolds.

Abstract

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on -bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and have Legendrian-isotopic components and (2) a pair of Legendrian links that are not Legendrian isotopic, but are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and which are link-homotopic as Legendrian links. Moreover, we construct examples showing that both of these non-simplicity phenomena can occur in the same smooth isotopy class. To construct these examples, we develop the theory of links transverse to a nowhere-zero vector field in a 3-manifold, and construct analogous examples in the category of links transverse to a vector field.

Paper Structure

This paper contains 9 sections, 18 theorems, 10 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on $V$-transverse links
  3. Generalized Classical Invariants and the Botany Problem
  4. Kinks and stabilization
  5. Spaces of Framed and $V$-Transverse Multicurves
  6. Approximating by $V$-transverse homotopies and isotopies
  7. Generators of $\pi_1$ of the spaces of immersed, framed, and $V$-transverse curves in an $S^1$-bundle over a surface
  8. A Finite-Type Invariant of $V$-transverse links
  9. Examples

Key Result

Theorem 2

Let $M$ be an $S^1$-bundle over an oriented surface of genus at least 2. For each $k\in\mathbb{Z}$ we construct a nowhere-zero vectorfield $V_k$ on $M$, and $V_k$-transverse links in $(M, V_k)$, which are framed isotopic and homotopic as $V$-transverse multi-curves (i.e., have the same generalized c

Figures (9)

  • Figure 1: $V$-transverse stabilizations $K^i$ and $K^{-i}$. In this local chart, $\frac{\partial}{\partial z}$ points out of the page.
  • Figure 2: The Legendrian link component $L_{i,0}$ and $L_{0,i}$ in the front and Lagrangian projections. In the Lagrangian projection, $\frac{\partial}{\partial z}$ points out of the page.
  • Figure 3: positive and negative crossing.
  • Figure 4: Four different types of kinks. The coordinates $(a,b)$ represent their contribution $a$ to a local rotation number and $b$ to a local writhe number of the diagram. Pairs of kinks with opposite local rotation number and opposite local writhe number can be created or cancelled by a $V$-transverse isotopy as shown in CC.
  • Figure 5: A loop in the space of $V$-transverse immersions corresponding to the $S^1$-fiber of $E_V$, created by sliding a small kink around a knot until it returns to its starting point.
  • ...and 4 more figures

Theorems & Definitions (38)

  • Remark 1
  • Theorem 2
  • Corollary 3
  • Theorem 4: CC
  • Definition 1
  • Definition 2
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 28 more