从三维拓扑量子场论视角看卫星构造与环面分解
A TQFT perspective on satellite constructions and toral decompositions
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中文总结 AI 辅助
从三维拓扑量子场论视角考虑纽结理论的卫星和拼接构造，给出相关多线性算子规定及构建链补态的方法，该构造与JSJ分解相关，通过研究几种链补的纠缠熵说明不同JSJ结构导致不同纠缠模式。
中文摘要 AI 辅助
我们从三维拓扑量子场论（TQFT）中链补态的角度考虑纽结理论的经典卫星和拼接构造。这些态由TQFT在链补流形上制备，该流形是通过从封闭的三维流形中移除加厚链得到的具有多个不相交环面边界的流形。在此设定下，卫星和拼接构造通过沿环面边界胶合流形实现，TQFT为这些构建块分配作用于相应环面希尔伯特空间的多线性算子。我们给出这些算子的明确规定，并展示如何用它们构建链补态。此构造与链补流形的JSJ分解密切相关。拼接通过沿环面胶合构建流形，而JSJ分解沿本质环面将流形切成标准片。从TQFT角度看，这将链补态表示为与JSJ片相关的基本算子网络。我们通过研究塞弗特纤维化链补、霍普夫链补和怀特海德加倍霍普夫链补的纠缠熵来说明此框架，展示它们不同的JSJ结构如何导致不同的纠缠模式。
英文摘要
We consider the classical satellite and splice constructions of knot theory from the perspective of link-complement states in three-dimensional topological quantum field theory (TQFT). These states are prepared by a TQFT on a link-complement manifold, namely a manifold with multiple disjoint torus boundaries, obtained by removing a thickened link from a closed ambient three-manifold. In this setting, satellite and splice constructions are realized by gluing manifolds along torus boundaries, and the TQFT assigns to these building blocks multilinear operators acting on the corresponding torus Hilbert spaces. We give explicit prescriptions for these operators and show how they can be used to build link-complement states. This construction is closely related to the JSJ decomposition of link-complement manifolds. While splicing builds manifolds by gluing along tori, the JSJ decomposition cuts a manifold along essential tori into canonical pieces. From the TQFT point of view, this expresses a link-complement state as a network of elementary operators associated with the JSJ pieces. We illustrate this framework by studying the entanglement entropy of Seifert-fibered link complements, Hopf-chain complements, and complements of Whitehead-doubled Hopf chains, showing how their different JSJ structures lead to distinct entanglement patterns.
