Legendrian knots and open book decompositions

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2009

Öğrenci: SİNEM ÇELİK ONARAN

Danışman: MUSTAFA KORKMAZ

Özet:

In this thesis, 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 of a Legendrian knot L in a contact 3-manifold as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framings given by the contact structure and the page agree. For any topological link in 3-sphere we construct a planar open book decomposition whose monodromy is a product of positive Dehn twists such that the planar open book contains the link on its page. Using this, we show any topological link, in particular any knot in any 3-manifold M sits on a page of a planar open book decomposition of M and we show any null-homologous loose Legendrian knot in an overtwisted contact structure has support genus zero.