Kapalı kontakt 5-manifoldlarda gömülü kapalılegendrian yüzeylerin maksimum sayfa geçişme sayısı

The main purpose of this thesis is to introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$- manifold $(M, \xi )$ which admits an "admissable" open book $(B, f)$ (supporting $\xi$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}_{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}_{(B,f)}(L)$, called "Absolute Maximal Page Crossing Number of $L$ with respect to $(B, f)$", which is invariant under Legendrian isotopies of $L$. In particular, this new invariant distinguishes Legendrian surfaces in the standard five-sphere which can not be distinguished by Thurston-Bennequin invariant. We give definitions of $M\mathcal{P}_{X}(L)$ and $M\mathcal{P}_{(B,f)}(L)$ and show that the invariants are well defined. Also, we show that they are preserved under Legendrian isotopies of $L$. Finally, we give an example about these invariants.