Asymptotics of the quantum invariants for surgeries on the figure 8 knot
Asymptotics of the quantum invariants for surgeries on the figure 8 knot
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DOI：
10.1142/s0218216506004555
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发表时间：
2006-04-01
影响因子：
0.5
通讯作者：
Hansen, SK
中科院分区：
文献类型：
--
作者：
Andersen, JE;Hansen, SK
我们研究了通过沿着8字形纽结做任何有理手术得到的3-流形M的与SU（2）相关的Reshetikhin-Turaev不变量。特别是，我们表示这些不变量在某些复杂的双重轮廓积分。这些积分公式使我们能够提出一个公式的领先渐近的不变量在大量子水平的限制。我们使用鞍点方法分析这个表达式。我们从相函数的驻点集合到M的基本群的非交换SL（2，C）-表示的共轭类空间上构造了一个满射，并证明了这些相函数在驻点处的值等于相应的平坦SU（2）-联络的经典Chern-Simons不变量.我们的研究结果与渐近展开猜想是一致的。此外，我们还计算了Kashaev [14]之后的8字形纽结的有色Jones多项式的领先渐近性。这导致了比体积猜想[24]预测的更精细的不变量渐近描述。
We investigate the Reshetikhin-Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2,C)-representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern-Simons invariants of the corresponding flat SU(2)-connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].