论文标题
相对Reshetikhin-turaev不变，双曲线锥体指标和离散的傅立叶变换i
Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms I
论文作者
论文摘要
我们提出了相对的猜想的猜想，该猜想是一个封闭式$ 3 $ manifold的相对reshetikhin-turaev不变的，其内部有彩色的框架链接，其渐近行为与数量相关，而Chern-simons则与Chern-simons相关，而Chern-Simons则是由单个链路和圆锥形的彩色磁场上的多重锥度指标，由链路和圆锥形的颜色确定。我们证明，如果环境$ 3 $ manifold是通过沿着基本阴影链接的某些组成部分进行的整体手术获得的，并且环境歧管中的链接的补充是基本的阴影链接补充，以获得足够的小圆锥角。加上Costantino和Thurston的结果，可以通过沿着合适的基本阴影链接的某些组件进行整体手术来获得所有紧凑的$ 3 $ 3 $ MANIFOLDS，可以获得chen-Yang的coshetikhin-Turaev Invariant sperveraliants of thepromentspypermanif的$ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3的可能方法。我们还介绍了一个拓扑操作（换对操作）的家族，该家族将封闭式$ 3 $ manifold的所有对以及具有同型型互补的框架链接连接起来，这些链接与相应的相对相对Reshetikhin-turaev Invariants进行了部分离散的傅立叶转换。作为应用程序，我们为离散的傅立叶变换找到了一个泊松求和公式。
We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented $3$-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient $3$-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston's result that all compact oriented $3$-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen-Yang's Volume Conjecture for the Reshetikhin-Turaev invariants of closed oriented hyperbolic $3$-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented $3$-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin-Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.
