Invariants of Legendrian Knots from Open Book Decompositions
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, no.10, pp.1831-1859, 2010 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Publication Date: 2010
- Doi Number: 10.1093/imrn/rnp185
- Journal Name: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Page Numbers: pp.1831-1859
- Open Archive Collection: AVESIS Open Access Collection
- Middle East Technical University Affiliated: Yes
Abstract
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, 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 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.