Title: From disc patterns in the plane to character varieties of knot groups Show Chinese title

Title: 从平面中的圆盘图案到纽结群的特征簇

Abstract: Motivated by an experimental study of groups generated by reflections in planar patterns of tangent circles, we describe some methods for constructing and studying representation spaces of holonomy groups of infinite volume hyperbolic $3$-manifolds that arise from unknotting tunnels of links. We include full descriptions of our computational methods, which were guided by simplicity and generality rather than by being particularly efficient in special cases. This makes them easy for non-experts to understand and implement to produce visualisations that can suggest conjectures and support algebraic calculations in the character variety. Throughout, we have tried to make the exposition clear and understandable for graduate students in geometric topology and related fields. Show Chinese abstract

Abstract: 受对平面相切圆图案中反射生成群的实验研究的启发，我们描述了一些构建和研究无限体积双曲$3$-流形的holonomy群表示空间的方法，这些流形来自于链环的解结隧道。 我们完整地描述了我们的计算方法，这些方法是基于简单性和普遍性，而不是特别高效于特殊情况。 这使得非专家也容易理解和实现，以生成可以提出猜想并支持特征簇中代数计算的可视化结果。 throughout, 我们努力使阐述清晰易懂，适合几何拓扑及相关领域的研究生。