For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding $\mathfrak {gl}_2$ skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of $\mathfrak {gl}_2$ foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the $\mathfrak {gl}_2$ skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated $\mathfrak {gl}_2$ skein algebra and verify an analogue of a positivity conjecture of Fock and Goncharov and Thurston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman–Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda–Przytycki–Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them.

中文翻译：

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Khovanov同源性和骨架模块的分类

对于有限类型的每个定向曲面，我们为曲面的增厚部分中的链接构造一个函数Khovanov同源性，它采用对应的$ \ mathfrak {gl} _2 $绞链模块分类的值。后者是对Kauffman括号绞线代数的适度改进，其分类使用$ \ mathfrak {gl} _2 $泡沫类别进行构建，该类别允许进行有趣的非负分级。我们期望$ \ mathfrak {gl} _2 $ skein模块上的自然代数结构可以通过张量积来分类，该张量积使表面链接同源性仿函数成为单曲面。我们在目标类别上构造一个候选双子函数，并猜想它会扩展为单曲面结构。这将产生相关的$ \ mathfrak {gl} _2 $丝球代数的规范基础，并验证Fock和Goncharov和Thurston的阳性猜想的类似物。我们通过检查圆环的绞球代数的Frohman-Gelca分类公式的几个实例，为单调猜想提供了证据。最后，我们恢复了Asaeda–Przytycki–Sikora表面链接同调的一种变体，并证明了表面嵌入在它们之间产生了光谱序列。