For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology.

中文翻译：

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枕套的深谷类、无痕性状变体、Khovanov 上同调

对于 3 球中的 2 股缠结图，我们在枕套光滑轨迹的 Fukaya 类别的三角包络中定义了紧致拉格朗日的扭曲复形。我们证明了这个扭曲的复合体是缠结同位素类的函子不变量，并且它提供了从缠结协同范畴到链状复合体的 Bar-Natan 函子的因式分解。特别是，我们的不变量的 hom 集具有与琐碎缠结相关的特定非紧致拉格朗日量，自然同构于缠结闭合的简化 Khovanov 链复合体。我们的构造来自与缠结图分辨率相关的无痕 SU(2) 字符变体的几何形状，并受到 Kronheimer 和 Mrowka 的奇异瞬时链接同源性的启发。