三维流形的量子迹映射和一个“长度猜想”
Quantum trace map for 3-manifolds and a 'length conjecture'
摘要：
我们引入了理想三角化双曲纽结补$S^3\backslash \mathcal{K}$的量子迹映射。该映射为3-流形的Kauffmann Skein模的每个元素分配一个量子算子。量子算子存在于理想三角剖分的量子化边参数模由胶合方程确定的一些等价关系的乘积所生成的模中。将量子映射与$SL(2,\mathbb{C})$ Chern-Simons理论的态积分模型相结合，可以定义纽结$K$在纽结补中的微扰不变量，纽结补的前导部分由其复双曲长度决定.然后，我们推测，微扰不变量确定的琼斯多项式的渐近展开组成的链接的$\mathcal{K}$和$K$。我们提出了8字形纽结补的显式量子迹映射，并在数值和解析两方面证实了长度猜想的二阶渐近展开。
英文摘要：
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of $SL(2,\mathbb{C})$ Chern-Simons theory, one can define perturbative invariants of knot $K$ in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of $\mathcal{K}$ and $K$. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.
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