Neighbourhoods and isotopies of knots in contact 3-manifolds

TL;DR

This paper establishes a neighborhood theorem for knots in contact 3-manifolds and demonstrates that topologically isotopic Legendrian knots can be made Legendrian isotopic through stabilizations.

Contribution

It provides a new neighborhood theorem for knots in contact 3-manifolds and shows stabilization can relate Legendrian isotopic knots.

Findings

Neighborhood theorem for knots in contact 3-manifolds

Legendrian knots become isotopic after stabilizations

Advances understanding of Legendrian knot isotopy classes

Abstract

We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.