论文标题
3个manifolds和“长度猜想”的量子跟踪图
Quantum trace map for 3-manifolds and a 'length conjecture'
论文作者
论文摘要
我们引入了理想的三角双曲打结$ s^3 \ backslash \ Mathcal {k} $的量子跟踪图。该地图将量子运算符分配给3个manifold的Kauffmann绞线模块的每个元素。量子运算符生活在由理想三角剖分模量的量化边缘参数产生的模块中，某些由胶合方程确定的等效关系。将量子图与$ sl（2，\ mathbb {c}）$ Chern-simons理论的状态综合模型相结合，可以定义结中的打结$ k $的扰动不变性，其领先部分由其复杂的双曲线长度确定。然后，我们猜想扰动不变性决定了由$ \ Mathcal {K} $和$ K $组成的链接的琼斯多项式扩展。我们提出了图形八个结的显式量子痕迹图，并在数值和分析上确认长度的猜想到渐近扩展中的二阶。
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.
