Invariants of Legendrian Knots from Open Book Decompositions

Onaran, Sinem Celik
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.

Citation Formats
S. C. Onaran, “Invariants of Legendrian Knots from Open Book Decompositions,” INTERNATIONAL MATHEMATICS RESEARCH NOTICES, no. 10, pp. 1831–1859, 2010, Accessed: 00, 2020. [Online]. Available: