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 [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. 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.

中文翻译：

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2 桥结和链接的辛四边形和抛物线表示

在本文中，我们通过在康威图上确定弧着色向量来研究 2 桥连接的抛物线表示。我们使用的方法是将共轭四边形方程组转换为辛四边形方程组。在这种方法中，我们对每个 2 桥链接 [公式：参见文本] 有一个整数系数一元多项式 [公式：参见文本]，并且该多项式的每个零在 [公式：见文本]满足辛四分方程系统，它给出了[公式：见文本]的抛物线表示的明确公式。然后，我们解释这些弧形着色向量如何为我们提供复体积的封闭形式公式和表示的尖头形状。作为该方法的其他应用，