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

TL;DR

This paper constructs infinite families of non-simple links in overtwisted contact structures on 3-manifolds, demonstrating complex isotopy phenomena and developing new link theory involving transverse links and vector fields.

Contribution

It introduces new examples of non-simple Legendrian links and develops a theory of links transverse to vector fields in 3-manifolds.

Findings

Existence of non-simple Legendrian links with identical classical invariants

Construction of links that are isotopic as framed links but not Legendrian isotopic

Development of a new theory of links transverse to vector fields

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.