Invariants of Legendrian knots from open book decompositions

TL;DR

This paper introduces a new invariant for Legendrian knots based on open book decompositions, defining support genus and demonstrating that loose knots in overtwisted structures have support genus zero, with implications for knot theory in contact manifolds.

Contribution

It defines the support genus invariant for Legendrian knots and proves that loose knots in overtwisted contact structures have support genus zero, linking open book decompositions to Legendrian knot invariants.

Findings

Support genus of Legendrian knots is minimal genus of supporting open book page.

Any null-homologous loose knot in an overtwisted structure has support genus zero.

Any topological knot can be placed on a planar open book page of the manifold.

Abstract

In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, \xi) as the minimal genus of a page of an open book of M supporting the contact structure \xi such that L sits on a page and the framing given by the contact structure and by the page agree. We show any null-homologous loose knot in an overtwisted contact structure has support genus zero. To prove this, we observe any topological knot or link in any 3-manifold M sits on a page of a planar open book decomposition of M.