Legendrian ribbons in overtwisted contact structures

Baader, S.; Cieliebak, K.; Vogel, T.

Legendrian ribbons in overtwisted contact structures

Authors

Baader, S.

Cieliebak, K.

Vogel, T.

Description

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 $\sel(K,S)=-χ(S)$. 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.
6 pages, 3 figures