勒让德纽结与可构造层
Legendrian knots and constructible sheaves
摘要：
本文研究了以勒让德结为结尾的拉格朗日膜的无包裹类。我们的纽结存在于曲面的余切丛中的接触无穷远处，曲面上的可构造层范畴等价于曲面本身上的可构造层范畴。因此，我们的范畴可以被描述为可构造的层，其奇异支撑由纽结的前投影控制。利用Guillermou-Kashiwara-Schapira的一个定理证明了所得范畴在Legendrian同位素下是不变的，并猜想它等价于Chekanov-Eliashberg微分分次代数的表示范畴.我们还发现两个连接拓扑纽结理论。首先，在环上画一个正的辫状闭包，秩为n的对象的模空间映射到圆上的局部系统的空间。与常数层的前推上的权过滤相关联的谱序列的第二页是（n着色的）三阶Khovanov-Rozansky同调。其次，在平面上画一个正的辫子闭包，我们的模空间在q元有限域上的点数恢复了辫子闭包的HOMFLY多项式的最小系数。
英文摘要：
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.
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