A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of $K_n$ with no empty lens. In this setting we prove an upper bound of $3((n-4)!)$ on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite and upper bounded by $n!$.

中文翻译：

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无空透镜的$K_n$星简图中的最大交叉数

星形简单图是相邻边不相交的图。相比之下，两条独立边之间的交叉次数没有限制。当允许空透镜（由两个边引起的排列中的一个面以 2 循环为界）时，两个独立的边可以在星形简单图中任意多次交叉。我们考虑没有空镜头的 $K_n$ 星形简单图。在此设置中，我们证明了任何一对边之间的最大交叉次数的上限为 $3((n-4)!)$。因此，交叉的总数是有限的，上限为 $n!$。