Invariants of Legendrian Knots from Open Book Decompositions

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