Transverse links, open books and overtwisted manifolds

TL;DR

This paper demonstrates that transverse links can be represented as sub-bindings of open books in contact manifolds, introduces the support genus concept, and explores its relation to overtwisted disks and Legendrian approximations.

Contribution

It establishes the realization of transverse links as sub-bindings, defines support genus, and connects it to overtwisted disks and Legendrian links.

Findings

Support genus of a transverse knot is zero if an overtwisted disk is disjoint.

Transverse links can be realized as sub-bindings of open book decompositions.

Relationship between support genus and Legendrian approximation is established.

Abstract

We prove that transverse links in any contact manifold $(M,\xi)$ can be realized as a sub-binding of a compatible open book decomposition. We define the support genus of a transverse link and prove that the support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it. Next, we find a relationship between the support genus of a transverse link and its Legendrian approximation.