论文标题
2桥结和链接的符号拼图和抛物线表示
Symplectic quandles and parabolic representations of 2-bridge Knots and Links
论文作者
论文摘要
在本文中，我们通过在Conway图上添加弧形矢量来研究2桥链接的抛物线表示。我们使用的方法是将共轭量程系统转换为符号谐波方程的系统。在这种方法中，我们为每种2桥链接$ k $提供了整数系数多项式$ p_k（u）$，并且此多项式的每个零为$ k $的图表提供了一组弧形颜色向量，使符号quandle方程式满足了$ k $ K $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $。然后，我们解释了这些弧形载体如何为我们提供复杂体积的封闭形式公式和表示形式的尖头形状。作为这种方法的其他应用，我们显示了莱利多项式和痕量场的一些有趣的算术特性，还描述了在相应的Riley多项式的分裂性方面，对于2桥连接组之间存在表达的必要条件。
In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ for each 2-bridge link $K$, and each zero of this polynomial gives a set of arc coloring vectors on the diagram of $K$ satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of $K$. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.
