The Universal sl_2 Link Homology Theory
The Universal sl_2 Link Homology Theory
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通用 sl_2 链接同源理论
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Gad Naot
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文献类型：
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作者：
Gad Naot
我们探索了与 Khovanov (n=2) 链接同源理论的几何形式中的链接相关的复形，确定其确切的底层代数结构，并找到其链接同源函子的精确普遍性属性。我们提出了直接从这个通用复合体中提取所有已知链接同源理论的新方法，并通过指定复合体中保存的信息量来确定其作为链接不变量的相对强度。
我们通过找到一种复杂的同构来实现这些目标，该同构将复杂的同构简化为更简单的类别。我们介绍了一些工具和方法，包括以 4TU/S/T 关系为模的表面分类和属生成算子，并使用它们来探索几何复形与其底层代数结构之间的关系。我们确定了可用于创建链接同源性的通用拓扑量子场论（TQFT），并发现它比 Khovanov 先前报告的“更小”。我们发现新的同源理论，相对于已知的信息，其信息量受到控制。
使用我们的归约定理可以有效地计算通用复形。这使我们能够通过普遍复合体的视角探索链接同源理论的现象学方面，以解释和统一各种现象（例如扭转和厚度）。普遍理论还使我们能够陈述由其衍生的特定链接同源理论的结果。本文开发的方法可以与其他已知技术（例如链路同源谱序列）结合或用于Khovanov链路同源性的各种扩展（例如sl_3链路同源性）。
We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex.
We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few tools and methods, including surface classification modulo the 4TU/S/T relations and genus generating operators, and use them to explore the relation between the geometric complex and its underlying algebraic structure. We identify the universal topological quantum field theory (TQFT) that can be used to create link homology and find that it is ``smaller'' than what was previously reported by Khovanov. We find new homology theories that hold a controlled amount of information relative to the known ones.
The universal complex is computable efficiently using our reduction theorem. This allows us to explore the phenomenological aspects of link homology theory through the eyes of the universal complex in order to explain and unify various phenomena (such as torsion and thickness). The universal theory also enables us to state results regarding specific link homology theories derived from it. The methods developed in this thesis can be combined with other known techniques (such as link homology spectral sequences) or used in the various extensions of Khovanov link homology (such as sl_3 link homology).