Invariants of Legendrian knots from open book decompositions

Details Invariants of Legendrian knots from open book decompositions Invariants of Legendrian knots from open book decompositions

2009-05-14

arxiv.org/abs/0905.2238v2

Mathematics

Geometric Topology

20 pages, 20 figures, proof of Lemma 2.1 added, minor corrections and clarifications

Scientific paper

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.

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