闭切触5-流形中Legendrian曲面的最大页交叉数
Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds
摘要：
我们引入了一个新的勒让德同痕不变量不变量的任何封闭的可定向勒让德曲面$L$嵌入在一个封闭的接触$5$流形$(M, ξ)$，它允许一个“可接纳的”开放的书$(B, f)$（支持$ξ$）为$L$。我们表明，任何这样的$L$和固定的页面$X$，可以分配一个整数$M\mathcal{P}_{X}(L)$，称为“相对最大跨页数的$L$相对于$X$“，这是不变的勒让德同位素下的$L$。我们还表明，可以将其扩展到无页不变式，即，可以指定一个整数$M\mathcal{P}_{(B,f)}(L)$，称为“$L$相对于$(B, f)$的绝对最大页交叉数“，它在$L$的勒让德同位素下是不变的。特别地，这个新的不变量区分了标准五球面中的Legendrian曲面，而Thurston-Bennequin不变量不能区分Legendrian曲面。
英文摘要：
We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) 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.
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