摘要翻译：
本文的主要目的是通过在Khovanov复形上添加新的梯度来构造具有附加结构的节点的新不变量。下面给出的想法适用于虚拟结、封闭编织和其他一些附加结构结的情况。我们附加分级的来源可能是拓扑的或组合的；它对许多部分情况是公理的。作为副产品，这导致了一个复合体，在某些情况下，它与通常的Khovanov复合体和在其他一些情况下与Lee-Rasmussen复合体相一致（直到分级重整化）。我们将要构造的分级对于Khovanov同调的一些推广，例如Frobenius扩张，表现得很好。这些新的同调理论对一些结的特征，如最小交叉数、原子亏格、片亏格等给出了更清晰的估计。我们的梯度在通常的Khovanov复形上产生了自然的过滤。存在一个从我们的同调开始并收敛到通常的Khovanov同调的谱序列。
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英文标题：
《Additional Gradings in Khovanov Homology》
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作者：
Vassily Olegovich Manturov
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Geometric Topology        几何拓扑
分类描述：Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
流形，轨道，多面体，细胞复合体，叶状，几何结构
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be topological or combinatorial; it is axiomatised for many partial cases. As a byproduct, this leads to a complex which in some cases coincides (up to grading renormalisation) with the usual Khovanov complex and in some other cases with the Lee-Rasmussen complex.   The grading we are going to construct behaves well with respect to some generalisations of the Khovanov homology, e.g., Frobenius extensions. These new homology theories give sharper estimates for some knot characteristics, such as minimal crossing number, atom genus, slice genus, etc.   Our gradings generate a natural filtration on the usual Khovanov complex. There exists a spectral sequence starting with our homology and converging to the usual Khovanov homology.
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PDF链接：
https://arxiv.org/pdf/0710.3741