An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digon-free intersecting arrangements of 6 pseudocircles we identify the three non-circularizable examples. We also show non-circularizability of 8 additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our non-circularizability proofs depend on incidence theorems like Miquel's. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way. The claims that we have all non-circularizable arrangements with the given properties are based on a program that generated all arrangements up to a certain size. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed were examined by hand.

中文翻译：

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伪圆的排列：关于可循环性

伪圆的排列是球体上或平面上的简单闭合曲线的集合，这样任意两条曲线要么不相交，要么恰好在两个交叉点相交。如果每对伪圆相交两次，我们称这种排列为相交。如果存在组合等效的圆排列，则该排列是可环化的。在本文中，我们展示了首次全面研究可循环性的结果。我们证明了 5 个伪圆恰好有 4 个不可圆化的排列（其中一个是以前已知的）。在 6 个伪圆的 2131 个无对角相交排列的集合中，我们确定了三个不可圆化的例子。我们还展示了 6 个伪圆的 8 个额外排列的不可圆化性，这些伪圆具有一组大小至少为 4 的对称性。我们的大多数不可循环证明都依赖于像 Miquel 的关联定理。在其他情况下，我们通过考虑连续变形来反驳可循环性，其中假设的圆表示的圆以受控方式增长或收缩。声称我们拥有所有具有给定属性的不可循环排列是基于一个程序，该程序生成了特定大小的所有排列。给定完整的排列列表，我们使用启发式方法来找到圆形表示。手工检查启发式失败的示例。声称我们拥有所有具有给定属性的不可循环排列的说法是基于一个程序，该程序生成了特定大小的所有排列。给定完整的排列列表，我们使用启发式方法来找到圆形表示。手工检查启发式失败的示例。声称我们拥有所有具有给定属性的不可循环排列是基于一个程序，该程序生成了特定大小的所有排列。给定完整的排列列表，我们使用启发式方法来找到圆形表示。手工检查了启发式失败的示例。