The universal Khovanov link homology theory
The universal Khovanov link homology theory
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通用霍瓦诺夫链接同调理论
DOI：
10.2140/agt.2006.6.1863
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发表时间：
2006
影响因子：
0.7
通讯作者：
Gad Naot
中科院分区：
文献类型：
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作者：
Gad Naot
我们确定了与Khovanov链同调理论（nD 2）的Bar-Natan几何形式中的链相关的几何复形的代数结构。我们发现了一个同构的复杂性，减少复杂的一个在一个简单的类别。这种减少使我们能够准确地指定的信息量内的几何复杂，从而准确地说明其普适性的联系同源性理论。我们还确定其强度作为一个链接不变相对于不同的拓扑量子场论（TQFT）用于创建链接同源性。我们确定了最一般的（通用的）TQFT，可用于创建链接同源性，并发现它是“更小”比TQFT先前报道的Khovanov作为通用链接同源性理论。我们给出了一个新的方法提取所有其他链接同源理论（包括Khovanov的通用TQFT）直接从通用几何复杂，沿着新的同源理论，持有控制量的信息。我们实现了这些目标，通过分类的表面（边界）模的4 TU/S/T关系，一个过程中涉及到引进属生成算子。这些算子使我们能够探索几何复形与其代数结构之间的关系。
We determine the algebraic structure underlying the geometric complex associated to a link in Bar-Natan’s geometric formalism of Khovanov’s link homology theory (nD 2). We find an isomorphism of complexes which reduces the complex to one in a simpler category. This reduction enables us to specify exactly the amount of information held within the geometric complex and thus state precisely its universality properties for link homology theories. We also determine its strength as a link invariant relative to the different topological quantum field theories (TQFTs) used to create link homology. We identify the most general (universal) TQFT that can be used to create link homology and find that it is “smaller” than the TQFT previously reported by Khovanov as the universal link homology theory. We give a new method of extracting all other link homology theories (including Khovanov’s universal TQFT) directly from the universal geometric complex, along with new homology theories that hold a controlled amount of information. We achieve these goals by making a classification of surfaces (with boundaries) modulo the 4TU/S/T relations, a process involving the introduction of genus generating operators. These operators enable us to explore the relation between the geometric complex and its algebraic structure.