Additional Gradings in Khovanov Homology
Additional Gradings in Khovanov Homology
复制标题
Khovanov 同源性的附加分级
DOI：
10.1142/9789812819116_0013
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
V. Manturov
中科院分区：
文献类型：
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作者：
V. Manturov
本文的主要目标是通过在Khovanov复形上增加新的分次来构造具有附加结构的纽结的新的不变量。下面给出的想法适用于虚拟结、闭合编织物和具有附加结构的结的一些其他情况。我们的额外分级的来源可能是拓扑的或组合的，它是公理化的许多部分情况下。作为副产品，这导致了一个复杂的，在某些情况下，符合（分级重正化）与通常的Khovanov复杂，并在其他一些情况下与李-拉斯穆森复杂。
我们将要构造的分级对于Khovanov同调的一些概括表现良好，例如，Frobenius扩展。这些新的同调理论对纽结的一些性质，如最小交叉数、原子亏格、切片亏格等给出了更精确的估计。
我们的分级在通常的Khovanov复合体上产生自然过滤。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。
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.