Legendrian knots and open book decompositions

Download
2009
Çelik Onaran, Sinem
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.

Suggestions

Open book decompositions of links of quotient surface singularities
Yılmaz, Elif; Korkmaz, Mustafa; Department of Mathematics (2009)
In this thesis, we write explicitly the open book decompositions of links of quotient surface singularities that support the corresponding unique Milnor fillable contact structures. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the corresponding unique Milnor fillable contact structures. That minimal page-genus is called Milnor genus. In this thesis we also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singular...
On the algebraic structure of relative hamiltonian diffeomorphism group
Demir, Ali Sait; Ozan, Yıldıray; Department of Mathematics (2008)
Let M be smooth symplectic closed manifold and L a closed Lagrangian submanifold of M. It was shown by Ozan that Ham(M,L): the relative Hamiltonian diffeomorphisms on M fixing the Lagrangian submanifold L setwise is a subgroup which is equal to the kernel of the restriction of the flux homomorphism to the universal cover of the identity component of the relative symplectomorphisms. In this thesis we show that Ham(M,L) is a non-simple perfect group, by adopting a technique due to Thurston, Herman, and Banyag...
On the tight contact structures on seıfert fibred 3−manifolds with 4 singular fibers
Medetoğulları, Elif; Ozan, Yıldıray; Department of Mathematics (2010)
In this thesis, we study the classification problem of Stein fillable tight contact structures on any Seifert fibered 3−manifold M over S 2 with 4 singular fibers. In the case e0(M) · −4 we have a complete classification. In the case e0(M) ¸ 0 we have obtained upper and lower bounds for the number of Stein fillable contact structures on M.
Liftable homeomorphisms of cyclic and rank two finite abelian branched covers over the real projective plane
Atalan, Ferihe; Medetogullari, Elif; Ozan, Yıldıray (Elsevier BV, 2021-02-01)
© 2020 Elsevier B.V.In this note, we investigate the property for regular branched finite abelian covers of the real projective plane, where each homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface.
Invariants of Legendrian Knots from Open Book Decompositions
Onaran, Sinem Celik (Oxford University Press (OUP), 2010-01-01)
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 supp...
Citation Formats
S. Çelik Onaran, “Legendrian knots and open book decompositions,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.