2-有规定端点的球面上的弧系
2-systems of arcs on spheres with prescribed endpoints
摘要：
令$S$为$n$穿孔球体，其中具有$n \geq 3$。证明了$\binom{n}{3}$是$S$上的两两非同伦简单弧族的最大长度，该弧族连接$S$的一对固定的不同的穿孔，并且两两相交最多两次.在此过程中，我们证明了一个正方形环形图$A$有一个角的每一个边界路径上，如果$A$包含至少一个正方形和对偶曲线的$A$是简单的弧加入边界路径的$A$和成对相交最多一次。
英文摘要：
Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.
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