论文标题
Atiyah-Patodi-Singer Rho不变和链接签名
The Atiyah-Patodi-Singer rho invariant and signatures of links
论文作者
论文摘要
很长一段时间以来，Atiyah-Patodi-Singer Rho不变性与链接的签名之间的关系已经很长时间了，但仅部分研究了它们。为了进一步探索它们，我们为Rho不变性开发了一种多功能的剪切公式，这使我们能够以方便的方式操纵歧管。在此工具的帮助下，我们将链接$ l $的多变量签名描述为某些封闭的$ 3 $ -Manifold $ y_l $与$ L $内在相关的Rho不变性。然后，我们研究了Dehn手术在整数和理性框架沿$ L $获得的流形的Rho不变性。受卡森，戈登，西马索尼和佛罗伦萨的相关结果的启发，我们给出了表达该价值的公式，作为$ l $的多变量签名和一些易于计算的额外条款的总和。
Relations between the Atiyah-Patodi-Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed $3$-manifold $Y_L$ intrinsically associated to $L$. We study then the rho invariant of the manifolds obtained by Dehn surgery on a $L$ along integer and rational framings. Inspired by related results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms.
