Legendrian ribbons in overtwisted contact structures

TL;DR

This paper characterizes when null-homologous transverse knots in overtwisted contact 3-manifolds are boundaries of Legendrian ribbons, linking their self-linking number to the Euler characteristic of Seifert surfaces, and relates tightness to genus minimization.

Contribution

It provides a criterion for Legendrian ribbon boundaries in overtwisted contact structures and characterizes tightness via genus minimization of Legendrian ribbons.

Findings

A null-homologous transverse knot bounds a Legendrian ribbon iff its self-linking number equals minus the Euler characteristic of a Seifert surface.

Every null-homologous topological knot type in an overtwisted contact manifold can be realized as a Legendrian ribbon boundary.

A contact structure is tight iff every Legendrian ribbon minimizes genus in its relative homology class.

Abstract

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-\chi(S)$. In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.