反常滑轮与纽结接触同调
Perverse Sheaves and Knot Contact Homology
摘要：
本文给出了L意义下纽结接触同调的一个新的代数构造，这主要是一个研究性的公告。Ng [Ng05a]。对于$ {\mathbb R}^3 $中的链$L$，我们定义了一个具有多个对象的微分分次（DG）$k$范畴$ \tilde{\mathscr A} $，其拟等价类是$ L $的一个拓扑不变量.当$L$是一个纽结时，$ \tilde{\mathscr A} $的一个区别对象的自同态代数与Ekholm，Etnyre，Ng和Sullivan在[EENS 13 a]中定义的完全非交换纽结DGA一致。我们构造的输入是辫子群$B_n$在二维圆盘上的反常层范畴上的自然作用，该圆盘在$n$标记点处具有奇点，Gelfand，MacPherson和Vilonen在[GMV 96]中研究了这一点。作为一个应用，我们证明了定义为DG范畴$ \tilde{\mathscr A} $的$0$ th同调的链$k$ -范畴$ \tilde{A} = H_0(\tilde{\mathscr A}) $的有限维表示范畴等价于$ {\mathbb R}^3 $上沿链$ L $奇异的反常层范畴.通过推广Gelfand-MacPherson-Vilonen辫子作用，我们得到了范畴$ \tilde{\mathscr A} $的几个推广.
英文摘要：
In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in $ {\mathbb R}^3 $, we define a differential graded (DG) $k$-category $ \tilde{\mathscr A} $ with finitely many objects, whose quasi-equivalence class is a topological invariant of $ L $. In the case when $L$ is a knot, the endomorphism algebra of a distinguished object of $ \tilde{\mathscr A} $ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng and Sullivan in [EENS13a]. The input of our construction is a natural action of the braid group $B_n$ on the category of perverse sheaves on a two-dimensional disk with singularities at $n$ marked points, studied by Gelfand, MacPherson and Vilonen in [GMV96]. As an application, we show that the category of finite-dimensional representations of the link $k$-category $ \tilde{A} = H_0(\tilde{\mathscr A}) $ defined as the $0$th homology of our DG category $ \tilde{\mathscr A} $ is equivalent to the category of perverse sheaves on $ {\mathbb R}^3 $ which are singular along the link $ L $. We also obtain several generalizations of the category $ \tilde{\mathscr A} $ by extending the Gelfand-MacPherson-Vilonen braid action.
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