On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings
On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings
复制标题
6 交叉结的 Kashaev 不变量的渐近展开
DOI：
10.1017/s0305004117000494
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发表时间：
2017
影响因子：
0.8
通讯作者：
YOKOTA YOSHIYUKI
中科院分区：
文献类型：
--
作者：
OHTSUKI TOMOTADA;YOKOTA YOSHIYUKI
给出了具有6个交叉点的纽结的Kashaev不变量的渐近展开式。特别地，我们给出了这些纽结的体积猜想，它指出展开的先导项呈现双曲体积和纽结的补集的Chern-Simons不变量。作为展开式的高系数，我们得到了这些纽结的一系列新的不变量。证明的一个非平凡部分是应用鞍点方法计算一个积分的渐近展开式，该积分表示Kashaev不变量。这一部分的一个关键步骤是给出ℂ3中积分的(实三维)区域的一个具体同伦，使得该区域的边界总是驻留在ℂ3中由双曲结构的势函数给定的某个区域内。
We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.