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Maximal page crossing number of embedded closed legendrian surfaces in closed contact 5-manifolds

Erşen, Özlem
The main purpose of this thesis is to introduce a new Legendrian isotopy invariantfor any closed orientable Legendrian surfaceLembedded in a closed contact5- man-ifold(M,ξ)which admits an "admissable" open book(B,f)(supportingξ) forL.We show that to any suchLand a fixed pageX, one can assign an integerMPX(L),called "Relative Maximal Page Crossing Number ofLwith respect toX", which isinvariant under Legendrian isotopies ofL. We also show that one can extend thisto a page-free invariant, i.e., one can assign an integerMP(B,f)(L), called "Abso-lute Maximal Page Crossing Number ofLwith respect to(B,f)", which is invari-ant under Legendrian isotopies ofL. In particular, this new invariant distinguishesLegendrian surfaces in the standard five-sphere which can not be distinguished byThurston-Bennequin invariant.We give definitions ofMPX(L)andMP(B,f)(L)and show that the invariants arewell defined. Also, we show that they are preserved under Legendrian isotopies of L. Finally, we give an example about these invariants.