论文标题
抛物线表示和广义的莱利多项式
Parabolic representations and generalized Riley polynomials
论文作者
论文摘要
我们将R. Riley关于两个桥接结组的抛物线表示的研究概括为$ s^3 $的一般结。我们利用抛物线拼图方法来进行一般结图，并采用象征性的拼图进行更好的研究，从而明确提供了这种表示及其复杂的体积。对于具有指定交叉$ C $的任何结图，我们定义了\ Mathbb {q} [Q} [y] $的广义莱利多项式$ r_c（y）\，其根对应于结组的抛物面表示。抛物线搜索的签名类型是新引入的，我们获得了阻塞类的公式，以提升到边界单位$ \ text {sl} _2 \ mathbb {c} $ - 表示。此外，我们定义了另一个多项式$ g_c（u）\ in \ mathbb {q} [u] $，称为$ u $ -polynomial，并证明$ r_c（u^2）= \ pm g_c（u）g_c（u）g_c（u）g_c（-u）$。基于此结果，我们介绍并调查了与不变的痕迹字段密切相关的Riley Field和$ U $ Field。该方法最终导致结的抛物线群的完整分类及其复杂的体积和尖端形状，最多可达12个。
We generalize R. Riley's study about parabolic representations of two bridge knot groups to the general knots in $S^3$. We utilize the parabolic quandle method for general knot diagrams and adopt symplectic quandle for better investigation, which gives such representations and their complex volumes explicitly. For any knot diagram with a specified crossing $c$, we define a generalized Riley polynomial $R_c(y) \in \mathbb{Q}[y]$ whose roots correspond to the conjugacy classes of parabolic representations of the knot group. The sign-type of parabolic quandle is newly introduced and we obtain a formula for the obstruction class to lift to a boundary unipotent $\text{SL}_2 \mathbb{C}$-representation. Moreover, we define another polynomial $g_c(u)\in\mathbb{Q}[u]$, called $u$-polynomial, and prove that $R_c(u^2)=\pm g_c(u)g_c(-u)$. Based on this result, we introduce and investigate Riley field and $u$-field which are closely related to the invariant trace field. This method eventually leads to the complete classification of parabolic representations of knot groups along with their complex volumes and cusp shapes up to 12 crossings.
